reserve x,y,z,w for ExtReal,
  r for Real;
reserve f,g for ExtReal;

theorem Th46:
  0 < x & x < y implies 0 < y - x
proof
  assume that
A1: 0 < x and
A2: x < y;
A3: x in REAL by A1,A2,XXREAL_0:48;
A4: 0 <> y - x
  proof
    assume 0= y - x;
    then x = (y - x) + x by Th4
      .= y by A3,Th22;
    hence thesis by A2;
  end;
  0 + x < y by A2,Th4;
  hence thesis by A3,A4,Th45;
end;
