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 Th59:
  v in W implies a * v in v + W
proof
  assume v in W;
  then
A1: (a - 1) * v in W by Th21;
  a * v = ((a - 1) + 1) * v .= (a - 1) * v + 1 * v by RLVECT_1:def 6
    .= v + (a - 1) * v by RLVECT_1:def 8;
  hence thesis by A1;
end;
