theorem Th47:
  the carrier of W1 c= the carrier of W2 implies W1 is Subspace of W2
proof
  set VW1 = the carrier of W1;
  set VW2 = the carrier of W2;
  set AV = the addF of V;
  set MV = the Mult of V;
  assume
A1: the carrier of W1 c= the carrier of W2;
  then
A2: [:VW1,VW1:] c= [:VW2,VW2:] by ZFMISC_1:96;
  0.W1 = 0.V by Def8;
  hence the carrier of W1 c= the carrier of W2 & 0.W1 = 0.W2 by A1,Def8;
  the addF of W1 = AV||VW1 & the addF of W2 = AV||VW2 by Def8;
  hence the addF of W1 = (the addF of W2)||the carrier of W1 by A2,FUNCT_1:51;
A3: [:COMPLEX,VW1:] c= [:COMPLEX,VW2:] by A1,ZFMISC_1:95;
  the Mult of W1 = MV | [:COMPLEX,VW1:] & the Mult of W2 = MV | [:COMPLEX,
  VW2 :] by Def8;
  hence thesis by A3,FUNCT_1:51;
end;
