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

theorem Th96:
  y>=0 & z>=0 implies x*(y+z) = x*y +x*z
proof
  assume
A1: y>=0 & z>=0;
  per cases by A1;
  suppose
    y = 0 or z = 0;
    hence thesis by Lm25;
  end;
  suppose
A2: y>0 & z>0;
    per cases;
    suppose
      x is real;
      hence thesis by Th95;
    end;
    suppose
      x is not real;
      then
A3:   not x in REAL;
      per cases by A3,XXREAL_0:14;
      suppose
A4:     x = -infty;
        hence x*(y+z) = -infty by A2,Def5
          .= -infty + x*z by A2,A4,Def2
          .= x*y + x*z by A2,A4,Def5;
      end;
      suppose
A5:     x = +infty;
        hence x*(y+z) = +infty by A2,Def5
          .= +infty + x*z by A2,A5,Def2
          .= x*y + x*z by A2,A5,Def5;
      end;
    end;
  end;
end;
