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

theorem :: MEMBER_1:2
  (f*g)" = f"*g"
proof
  per cases by XXREAL_0:14;
  suppose
    f in REAL & g in REAL;
    then reconsider f1 = f, g1 = g as Real;
A1: (ex a being Complex st f1 = a & f" = a" )& ex b being Complex
 st g1 = b & g" = b" by Def6;
    then ex a,b being Complex st f" = a & g" = b & f"*g" = a * b by Def5;
    then f"*g" = (f1*g1)" by A1,XCMPLX_1:204;
    hence thesis;
  end;
  suppose
A2: f = +infty;
    g is positive or g is negative or g = 0;
    then (f*g)" = +infty" or (f*g)" = -infty" or (f*g)" = 0" by A2,Def5;
    hence thesis by A2;
  end;
  suppose
A3: f = -infty;
    g is positive or g is negative or g = 0;
    then (f*g)" = +infty" or (f*g)" = -infty" or (f*g)" = 0" by A3,Def5;
    hence thesis by A3;
  end;
  suppose
A4: g = +infty;
    f is positive or f is negative or f = 0;
    then (f*g)" = +infty" or (f*g)" = -infty" or (f*g)" = 0" by A4,Def5;
    hence thesis by A4;
  end;
  suppose
A5: g = -infty;
    f is positive or f is negative or f = 0;
    then (f*g)" = +infty" or (f*g)" = -infty" or (f*g)" = 0" by A5,Def5;
    hence thesis by A5;
  end;
end;
