prove that GDP equals Consumption + Saving + Taxes?

It is a definition of income in the Keynesian model in a closed economy.

y= c+s+t and

y=c+i+g

That means c+s+t=c+i+g

In equilibrium,i=s,then t=g.

The income after taxes will be used for consumption or saving.

The second equation means income=expenditure.

GDP is the sum total of all expenditures multiplied by net export of the country in question.

GDP = Consumer consumption + Government spending + Business spending (export-import).

It sums up the total of expenditures to show how money is going around in the economy. If all or any of these factors go up, it contributes to the overall GDP of the country.

Example, government spending brings a ripple effect on business spending (If the government build roads, it buys cement, steel etc, from businesses), and therefore increase company profits, but have to hire more workers, buy more raw materials, etc. Then when the company hires more workers, more people receives salaries, which they buy stuff they like, eat at restaurants, etc.

Same way when they increase their spending on raw materials, because owners of these raw material companies will have to jack up production, hire more workers.

When more workers are hired, they have spending money for shopping. Shopping increases demand in the market place, therefore, manufacturer's (business owners) have to buy more raw mat, etc.. and the cycle goes on.

Net export is Export minus Import. A country has to have more export than import to not affect the overall GDP.

This is true for closed economy with benevolent government.

In closed economy , NX =0 and so, S =I

A benevolent government ensures deficit/surplus =0, so G =T

The standard form Y = C +G +I +NX

returns

Y = C + S + T

IT does NOT

GDP has to be defined by listing what is included

what is the GDP contribution of a stay at home mother and (house) wife?

economics is NOT a "hard science "like physics even though it uses math formulas that are approximations and guess of economic behavior