reserve x,y,y1,y2 for object;
reserve R for Ring;
reserve a for Scalar of R;
reserve V,X,Y for RightMod of R;
reserve u,u1,u2,v,v1,v2 for Vector of V;
reserve V1,V2,V3 for Subset of V;
reserve W,W1,W2 for Submodule of V;
reserve w,w1,w2 for Vector of W;

theorem Th27:
  the carrier of W1 c= the carrier of W2 implies W1 is Submodule of W2
proof
  set VW1 = the carrier of W1;
  set VW2 = the carrier of W2;
  set MW1 = the rmult of W1;
  set MW2 = the rmult of W2;
  set AV = the addF of V;
  set MV = the rmult 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,the carrier of
  W1:] qua set) by A1,FUNCT_1:51;
A4: MW1 = MV |([:VW1,the carrier of R:] qua set) & MW2 = MV |([:VW2,the
  carrier of R:] qua set) by Def2;
  [:VW1,the carrier of R:] c= [:VW2,the carrier of R:] by A2,ZFMISC_1:95;
  then
A5: MW1 = MW2 |([:VW1,the carrier of R:] 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;
