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

theorem
  i1<=i2 implies i1-'1<=i2
proof
  assume
A1: i1<=i2;
  reconsider i1,i2 as Nat by TARSKI:1;
  per cases;
  suppose i1-1>=0;
    then i1-'1=i1-1 by XREAL_0:def 2;
    then i1-'1<=i1-1+1 by NAT_1:12;
    hence thesis by A1,XXREAL_0:2;
  end;
  suppose i1-1<0;
    hence thesis by XREAL_0:def 2;
  end;
end;
