
theorem str12:
for F being Field
for E1,E2 being FieldExtension of F
st E1 == E2 holds VecSp(E1,F) = VecSp(E2,F)
proof
let F be Field, E1,E2 be FieldExtension of F;
set V1 = VecSp(E1,F), V2 = VecSp(E2,F);
assume AS: E1 == E2;
A: the carrier of V1 = the carrier of E2 by AS,FIELD_4:def 6
                    .= the carrier of V2 by FIELD_4:def 6;
B: the addF of V1 = the addF of E2 by AS,FIELD_4:def 6
                 .= the addF of V2 by FIELD_4:def 6;
C: the ZeroF of V1 = 0.E1 by FIELD_4:def 6
                  .= 0.E2 by AS
                  .= the ZeroF of V2 by FIELD_4:def 6;
   the lmult of V1 = (the multF of E2)|[:the carrier of F,the carrier of E2:]
                     by AS,FIELD_4:def 6
                  .= the lmult of V2 by FIELD_4:def 6;
hence VecSp(E1,F) = VecSp(E2,F) by A,B,C;
end;
