
theorem
  for m,n be non zero Element of NAT,
        M be Matrix of m,n,F_Real
  holds
    Mx2Tran(M) is Lipschitzian LinearOperator
    of REAL-NS m,REAL-NS n
proof
  let m,n be non zero Element of NAT,
        M be Matrix of m,n, F_Real;

  the carrier of (TOP-REAL n) = the carrier of (REAL-NS n)
  & the carrier of (TOP-REAL m) = the carrier of (REAL-NS m) by REAL_NS2:4;

  then reconsider f = Mx2Tran(M) as Function of REAL-NS m,REAL-NS n;

  for x, y be Element of REAL-NS m
  holds f.(x + y) = f.x + f.y
  proof
    let x, y be Element of REAL-NS m;
    reconsider x0 = x, y0 = y as Element of TOP-REAL m by REAL_NS2:4;

    thus f.(x + y)
      = (Mx2Tran(M)).(x0 + y0) by REAL_NS2:7
    .= (Mx2Tran(M)).x0 + (Mx2Tran(M)).y0 by MATRTOP1:22
    .= f.x + f.y by REAL_NS2:7;
  end;
  then
  A1: f is additive;

  for x be VECTOR of REAL-NS m, a be Real
  holds f.(a * x) = a * f.x
  proof
    let x be VECTOR of REAL-NS m, a be Real;
    reconsider x0 = x as Element of TOP-REAL m by REAL_NS2:4;

    thus f.(a * x)
      = (Mx2Tran(M)).(a * x0) by REAL_NS2:8
    .= a * (Mx2Tran(M)).x0 by MATRTOP1:23
    .= a * f.x by REAL_NS2:8;
  end;
  then reconsider f as LinearOperator of REAL-NS m,REAL-NS n
    by A1,LOPBAN_1:def 5;

    REAL-NS m is finite-dimensional
  & dim(REAL-NS m) = m by REAL_NS2:62;
  then f is Lipschitzian by LOPBAN15:2;
  hence thesis;
end;
