theorem Th9:
  F is_left_distributive_wrt G implies F/\/RD is_left_distributive_wrt G/\/RD
proof
  deffunc Cl(Element of D) = EqClass(RD,$1);
  defpred P[Element of Class RD, Element of Class RD, Element of Class RD]
means (F/\/RD).($1,(G/\/RD).($2,$3)) = (G/\/RD).((F/\/RD).($1,$2),(F/\/RD).($1,
  $3));
  assume
A1: for d,a,b holds F.(d,G.(a,b)) = G.(F.(d,a),F.(d,b));
A2: now
    let x1,x2,x3 be Element of D;
    (F/\/RD).(Cl(x1),(G/\/RD).(Cl(x2),Cl(x3))) = (F/\/RD).(Cl(x1),Cl(G.(x2
    ,x3))) by Th3
      .= Cl(F.(x1,G.(x2,x3))) by Th3
      .= Cl(G.(F.(x1,x2),F.(x1,x3))) by A1
      .= (G/\/RD).(Cl(F.(x1,x2)),Cl(F.(x1,x3))) by Th3
      .= (G/\/RD).((F/\/RD).(Cl(x1),Cl(x2)),Cl(F.(x1,x3))) by Th3
      .= (G/\/RD).((F/\/RD).(Cl(x1),Cl(x2)),(F/\/RD).(Cl(x1),Cl(x3))) by Th3;
    hence P[EqClass(RD,x1),EqClass(RD,x2),EqClass(RD,x3)];
  end;
  thus for c1,c2,c3 being Element of Class RD holds P[c1,c2,c3] from SchAux3(
  A2);
end;
