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

theorem Th42:
  i1<=i2 implies i1-'n<=i2-'n
proof
  assume
A1: i1<=i2;
  per cases;
  suppose i1-n>=0;
    then
A2: i1-'n=i1-n by XREAL_0:def 2;
    i1-n<=i2-n by A1,XREAL_1:9;
    hence thesis by A2,XREAL_0:def 2;
  end;
  suppose i1-n<0;
    hence thesis by XREAL_0:def 2;
  end;
end;
