V=R2I.∀a=(x1,y1)∈V,b=(x2,y2)∈Va+b=(x1,y1)+(x2,y2)=(0,y1+y2)∈VII.∀c∈R,∀a=(x1,y1)∈Vc⋅a=c⋅(x1,y1)=(0,cy1)∈V
Axioms
1)a+b=b+aa+b=(x1,y1)+(x2,y2)=(0,y1+y2)b+a=(x2,y2)+(x1,y1)=(0,y2+y1)==(0,y1+y2)=a+btrue
2.∀d=(x3,y3)∈V(a+b)+d=a+(b+d)a+(b+d)=(x1,y1)+(0,y2+y3)==(0,y1+y2+y3)(a+b)+d=(0,y1+y2)+(x3,y3)==(0,y1+y2+y3)=a+(b+c)true
3.∃0=(x2,y2)∈V,∀a=(x1,y1)∈Va+0=aa+0=(x1,y1)+(x2,y2)=(0,y1+y2)=a∃0false
4.∀a=(x1,y1)∈V∃b=(x2,y2)∈Va+b=0∃0false
5.∀c,k∈R,∀a=(x1,y1)∈V(c+d)⋅a=c⋅a+d⋅a(c+d)⋅a=(c+d)⋅(x1,y1)=(0,(c+d)y1)c⋅a=c⋅(x1,y1)=(0,cy1)d⋅a=d⋅(x1,y1)=(0,dy1)c⋅a+d⋅a=(0,cy1)+(0,dy1)=(0,cy1+dy1)==(c+d)⋅atrue
6.∀c∈R,∀a=(x1,y1)∈V,b=(x2,y2)∈Vc⋅(a+b)=c⋅a+c⋅bc⋅(a+b)=c⋅(0,y1+y2)=(0,c(y1+y2))c⋅a=(0,cy1)c⋅b=(0,cy2)c⋅a+c⋅b=(0,cy1)+(0,cy2)=(0,cy1+cy2)==c⋅(a+b)true
7.∀c,k∈R,∀a=(x1,y1)∈V(c⋅d)⋅a=c⋅(d⋅a)(c⋅d)⋅a=(c⋅d)⋅(x1,y1)=(0,c⋅dy1)d⋅a=d⋅(x1,y1)=(0,dy1)c⋅(d⋅a)=c⋅(0,dy1)=(0,cdy1)=(c⋅d)⋅atrue
8.∃1∈R,∀a=(x1,y1)∈V1⋅a=a1⋅a=1⋅(x1,y1)=(0,1⋅y1)=(0,y1)=afalse
V is not the subset a vector of R2
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