reserve x,y,y1,y2 for object;
reserve GF for add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr,
  V,X,Y for Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital non
  empty ModuleStr over GF;
reserve a for Element of GF;
reserve u,u1,u2,v,v1,v2 for Element of V;
reserve W,W1,W2 for Subspace of V;
reserve V1 for Subset of V;
reserve w,w1,w2 for Element of W;

theorem Th27:
  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 MW1 = the lmult of W1;
  set MW2 = the lmult of W2;
  set AV = the addF of V;
  set MV = the lmult of V;
A1: (the addF of W1) = AV||VW1 & (the addF of W2) = AV||VW2 by Def2;
  assume
A2: the carrier of W1 c= the carrier of W2;
  then [:VW1,VW1:] c= [:VW2,VW2:] by ZFMISC_1:96;
  then
A3: the addF of W1 = (the addF of W2)||the carrier of W1 by A1,FUNCT_1:51;
A4: MW1 = MV |([:the carrier of GF,VW1:] qua set) & MW2 = MV |([:the carrier
  of GF,VW2:] qua set) by Def2;
  [:the carrier of GF,VW1:] c= [:the carrier of GF,VW2:] by A2,ZFMISC_1:95;
  then
A5: MW1 = MW2 |([:the carrier of GF,VW1:] qua set) by A4,FUNCT_1:51;
  0.W1 = 0.V & 0.W2 = 0.V by Def2;
  hence thesis by A2,A3,A5,Def2;
end;
