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;
