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

theorem Th72:
  x < y & 0 < z & z <> +infty implies x*z < y*z
proof
  assume that
A1: x < y and
A2: 0 < z and
A3: z <> +infty;
A4: now
A5: x <> +infty & y <> -infty by A1,XXREAL_0:3,5;
    assume
A6: x*z = y*z;
    per cases by A5,XXREAL_0:14;
    suppose
A7:   x in REAL & y in REAL;
      z in REAL by A2,A3,XXREAL_0:14;
      then reconsider x,y,z as Real by A7;
      x*z < y*z by A1,A2,XREAL_1:68;
      hence contradiction by A6;
    end;
    suppose
A8:   y = +infty;
      then y*z = +infty by A2,Def5;
      then x = +infty or x = -infty by A2,A3,A6,Th69;
      hence contradiction by A1,A2,A6,A8;
    end;
    suppose
A9:   x = -infty;
      then x*z = -infty by A2,Def5;
      then y = +infty or y = -infty by A2,A3,A6,Th70;
      hence contradiction by A1,A2,A6,A9;
    end;
  end;
  x*z <= y*z by A1,A2,Th71;
  hence thesis by A4,XXREAL_0:1;
end;
