reserve a,b,p,k,l,m,n,s,h,i,j,t,i1,i2 for natural Number;

theorem
  i < j & k < j implies i -' k < j -' k
proof
  assume that
A1: i < j and
A2: k < j;
  per cases;
  suppose k <= i;
    then
A3: i -' k = i - k by XREAL_1:233;
    j -' k = j - k by A2,XREAL_1:233;
    hence thesis by A1,A3,XREAL_1:9;
  end;
  suppose k > i;
    then i - k < 0 by XREAL_1:49;
    then
A4: i -' k = 0 by XREAL_0:def 2;
    j -' k <> 0 by A2,Th36;
    hence thesis by A4;
  end;
end;
