reserve a,b,c,d for Real;
reserve r,s for Real;

theorem
  a <= b implies a-1 < b
proof
  assume a<=b;
  then
A1: a-1<=b-1 by Lm7;
  b-1<b-1+1 by Lm9;
  hence thesis by A1,XXREAL_0:2;
end;
