let S be set ; :: according to VECTSP12:def 3 :: thesis: ( S in rng <*G,F*> implies S is VectSp of K )

assume S in rng <*G,F*> ; :: thesis: S is VectSp of K

then consider i being object such that

A1: ( i in dom <*G,F*> & <*G,F*> . i = S ) by FUNCT_1:def 3;

dom <*G,F*> = {1,2} by FINSEQ_1:2, FINSEQ_1:89;

then ( i = 1 or i = 2 ) by A1, TARSKI:def 2;

hence S is VectSp of K by A1, FINSEQ_1:44; :: thesis: verum

assume S in rng <*G,F*> ; :: thesis: S is VectSp of K

then consider i being object such that

A1: ( i in dom <*G,F*> & <*G,F*> . i = S ) by FUNCT_1:def 3;

dom <*G,F*> = {1,2} by FINSEQ_1:2, FINSEQ_1:89;

then ( i = 1 or i = 2 ) by A1, TARSKI:def 2;

hence S is VectSp of K by A1, FINSEQ_1:44; :: thesis: verum