reserve A,B for Ordinal,
  K,M,N for Cardinal,
  x,x1,x2,y,y1,y2,z,u for object,X,Y,Z,X1,X2, Y1,Y2 for set,
  f,g for Function;
reserve m,n for Nat;

theorem Th36:
  n*m = n *^ m
proof
  defpred P[Nat] means $1*m = $1 *^ m;
A1: for n st P[n] holds P[n+1]
  proof
    let n such that
A2: P[n];
    thus (n+1)*m = n*m+1*m .= n *^ m +^ m by A2,Th35
      .= n *^ m +^ 1 *^ m by ORDINAL2:39
      .= ( n +^ 1) *^ m by ORDINAL3:46
      .= (succ Segm n) *^ m by ORDINAL2:31
      .= Segm (n+1) *^ m by NAT_1:38
      .= (n+1) *^ m;
  end;
A3: P[0] by ORDINAL2:35;
  for n holds P[n] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
