reserve r,s,t,u for Real;

theorem Th48:
  for X being LinearTopSpace, r being non zero Real holds mlt(r,X)
  " = mlt(r",X)
proof
  let X be LinearTopSpace, r be non zero Real;
A1: rng mlt(r,X) = [#]X by Th47;
  now
    let x be Point of X;
    consider u being object such that
A2: u in dom mlt(r,X) and
A3: x = mlt(r,X).u by A1,FUNCT_1:def 3;
    reconsider u as Point of X by A2;
A4: x = r*u by A3,Def13;
    mlt(r,X) is onto one-to-one by A1,Lm13,FUNCT_2:def 3;
    hence mlt(r,X)".x = (mlt(r,X) qua Function)".x by TOPS_2:def 4
      .= u by A3,Lm13,FUNCT_2:26
      .= 1*u by RLVECT_1:def 8
      .= (r*r")*u by XCMPLX_0:def 7
      .= r"*x by A4,RLVECT_1:def 7
      .= mlt(r",X).x by Def13;
  end;
  hence thesis by FUNCT_2:63;
end;
