set adds = the addF of V | [:(a * V),(a * V):];

[:(a * V),(a * V):] c= [: the carrier of V, the carrier of V:] by ZFMISC_1:96;

then [:(a * V),(a * V):] c= dom the addF of V by FUNCT_2:def 1;

then A1: dom ( the addF of V | [:(a * V),(a * V):]) = [:(a * V),(a * V):] by RELAT_1:62;

for z being object st z in [:(a * V),(a * V):] holds

( the addF of V | [:(a * V),(a * V):]) . z in a * V

[:(a * V),(a * V):] c= [: the carrier of V, the carrier of V:] by ZFMISC_1:96;

then [:(a * V),(a * V):] c= dom the addF of V by FUNCT_2:def 1;

then A1: dom ( the addF of V | [:(a * V),(a * V):]) = [:(a * V),(a * V):] by RELAT_1:62;

for z being object st z in [:(a * V),(a * V):] holds

( the addF of V | [:(a * V),(a * V):]) . z in a * V

proof

hence
the addF of V | [:(a * V),(a * V):] is Function of [:(a * V),(a * V):],(a * V)
by A1, FUNCT_2:3; :: thesis: verum
let z be object ; :: thesis: ( z in [:(a * V),(a * V):] implies ( the addF of V | [:(a * V),(a * V):]) . z in a * V )

assume A2: z in [:(a * V),(a * V):] ; :: thesis: ( the addF of V | [:(a * V),(a * V):]) . z in a * V

consider x, y being object such that

A3: ( x in a * V & y in a * V & z = [x,y] ) by A2, ZFMISC_1:def 2;

consider v being Element of V such that

A4: x = a * v by A3;

consider w being Element of V such that

A5: y = a * w by A3;

( the addF of V | [:(a * V),(a * V):]) . z = (a * v) + (a * w) by A2, A3, A4, A5, FUNCT_1:49

.= a * (v + w) by VECTSP_1:def 14 ;

hence ( the addF of V | [:(a * V),(a * V):]) . z in a * V ; :: thesis: verum

end;assume A2: z in [:(a * V),(a * V):] ; :: thesis: ( the addF of V | [:(a * V),(a * V):]) . z in a * V

consider x, y being object such that

A3: ( x in a * V & y in a * V & z = [x,y] ) by A2, ZFMISC_1:def 2;

consider v being Element of V such that

A4: x = a * v by A3;

consider w being Element of V such that

A5: y = a * w by A3;

( the addF of V | [:(a * V),(a * V):]) . z = (a * v) + (a * w) by A2, A3, A4, A5, FUNCT_1:49

.= a * (v + w) by VECTSP_1:def 14 ;

hence ( the addF of V | [:(a * V),(a * V):]) . z in a * V ; :: thesis: verum