reserve V,X,Y for RealLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve a for Real;
reserve V1,V2,V3 for Subset of V;
reserve x for object;
reserve W,W1,W2 for Subspace of V;
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 [:REAL,D:],D;
reserve B,C for Coset of W;

theorem Th58:
  a <> 1 & a * v in v + W implies v in W
proof
  assume that
A1: a <> 1 and
A2: a * v in v + W;
A3: a - 1 <> 0 by A1;
  consider u such that
A4: a * v = v + u and
A5: u in W by A2;
  u = u + 0.V
    .= u + (v - v) by RLVECT_1:15
    .= a * v - v by A4,RLVECT_1:def 3
    .= a * v - 1 * v by RLVECT_1:def 8
    .= (a - 1) * v by RLVECT_1:35;
  then (a - 1)" * u = ((a - 1)" * (a - 1)) * v by RLVECT_1:def 7;
  then 1 * v = (a - 1)" * u by A3,XCMPLX_0:def 7;
  then v = (a - 1)" * u by RLVECT_1:def 8;
  hence thesis by A5,Th21;
end;
