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;
reserve B,C for Coset of W;

theorem Th69:
  z <> 0 & (z * v) + W = the carrier of W implies v in W
proof
  assume that
A1: z <> 0 and
A2: (z * v) + W = the carrier of W;
  assume not v in W;
  then not 1r * v in W by Def5;
  then not (z" * z) * v in W by A1,XCMPLX_0:def 7;
  then not z" * (z * v) in W by Def4;
  then
A3: not z * v in W by Th40;
  0.V in W & z * v + 0.V = z * v by Th36,RLVECT_1:4;
  then z * v in {z * v + u : u in W};
  hence contradiction by A2,A3;
end;
