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

theorem
  -infty <> y & y <> +infty & y <> 0 implies x * y / y = x & x * (y / y) = x
proof
  assume that
A1: -infty <> y and
A2: y <> +infty and
A3: y <> 0;
  reconsider b = y as Element of REAL by A1,A2,XXREAL_0:14;
A4: x * y / y = x
  proof
    per cases;
    suppose
A5:   x = +infty;
      per cases by A3;
      suppose
A6:     0 < y;
        then x * y = +infty by A5,Def5;
        hence thesis by A2,A5,A6,Th83;
      end;
      suppose
A7:     y < 0;
        then x * y = -infty by A5,Def5;
        hence thesis by A1,A5,A7,Th84;
      end;
    end;
    suppose
A8:   x = -infty;
      per cases by A3;
      suppose
A9:     0 <y;
        then x * y = -infty by A8,Def5;
        hence thesis by A2,A8,A9,Th86;
      end;
      suppose
A10:    y < 0;
        then x * y = +infty by A8,Def5;
        hence thesis by A1,A8,A10,Th85;
      end;
    end;
    suppose
      x <> +infty & x <> -infty;
      then x in REAL by XXREAL_0:14;
      then reconsider a = x as Real;
      (x * y)/y = (a * b)/b .= a by A3,XCMPLX_1:89;
      hence thesis;
    end;
  end;
  y / y = 1 by A1,A2,A3,Th78;
  hence thesis by A4,Th81;
end;
