
theorem Th5:
  for m,n,x,y being Nat st n = (m-'x)+y holds m-'n <= x & n-'m <= y
  proof
    let m,n,x,y be Nat such that
A0: n = (m-'x)+y;
    per cases;
    suppose m <= n;
      then
A1:   m-'n = 0 & n-'m = n-m by NAT_2:8,XREAL_1:233;
      n <= m+y by A0,XREAL_1:6,NAT_D:35;
      hence thesis by A1,XREAL_1:20;
    end;
    suppose
      m > n;
      then
A2:   n-'m = 0 & m-'(m-'n) = n >= m-'x by A0,NAT_D:58,NAT_1:11,NAT_2:8;
      m-'(m-'n) = m-(m-'n) & m-'x >= m-x
        by XREAL_0:def 2,XREAL_1:233,NAT_D:35;
      then m-(m-'n) >= m-x by A2,XXREAL_0:2;
      hence thesis by XREAL_1:10,A2;
    end;
  end;
