reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem Th4:
  for r,s be Real st 0<=r holds max+(r*s) = r*(max+ s)
proof
  let r,s be Real;
  assume
A1: 0<=r;
  now
    per cases;
    case
A2:   0<=s;
      then max+(r*s) = r*s by A1,XXREAL_0:def 10;
      hence thesis by A2,XXREAL_0:def 10;
    end;
    case
A3:   s<0;
      then max+ s= 0 by XXREAL_0:def 10;
      hence thesis by A1,A3,XXREAL_0:def 10;
    end;
  end;
  hence thesis;
end;
