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 Th77:
  z <> 1r & z * v in v + W implies v in W
proof
  assume that
A1: z <> 1r and
A2: z * v in v + W;
A3: z - 1r <> 0 by A1;
  consider u such that
A4: z * v = v + u and
A5: u in W by A2;
  u = u + 0.V by RLVECT_1:4
    .= u + (v - v) by RLVECT_1:15
    .= z * v - v by A4,RLVECT_1:def 3
    .= z * v - 1r * v by Def5
    .= (z - 1r) * v by Th10;
  then (z - 1r)" * u = ((z - 1r)" * (z - 1r)) * v by Def4;
  then 1r * v = (z - 1r)" * u by A3,XCMPLX_0:def 7;
  then v = (z - 1r)" * u by Def5;
  hence thesis by A5,Th40;
end;
