reserve a,b for Complex;
reserve V,X,Y for ComplexLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve z,z1,z2 for Complex;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Subspace of V;
reserve x for set;
reserve w,w1,w2 for VECTOR of W;
reserve D for non empty set;
reserve d1 for Element of D;
reserve A for BinOp of D;
reserve M for Function of [:COMPLEX,D:],D;

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;
