theorem Th10:
  F is_right_distributive_wrt G implies F/\/RD is_right_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).((G/\/RD).($1,$2),$3) = (G/\/RD).((F/\/RD).($1,$3),(F/\/RD).($2,
  $3));
  assume
A1: for a,b,d holds F.(G.(a,b),d) = G.(F.(a,d),F.(b,d));
A2: now
    let x2,x3,x1 be Element of D;
    (F/\/RD).((G/\/RD).(Cl(x2),Cl(x3)),Cl(x1)) = (F/\/RD).(Cl(G.(x2,x3)),
    Cl(x1)) by Th3
      .= Cl(F.(G.(x2,x3),x1)) by Th3
      .= Cl(G.(F.(x2,x1),F.(x3,x1))) by A1
      .= (G/\/RD).(Cl(F.(x2,x1)),Cl(F.(x3,x1))) by Th3
      .= (G/\/RD).((F/\/RD).(Cl(x2),Cl(x1)),Cl(F.(x3,x1))) by Th3
      .= (G/\/RD).((F/\/RD).(Cl(x2),Cl(x1)),(F/\/RD).(Cl(x3),Cl(x1))) by Th3;
    hence P[EqClass(RD,x2),EqClass(RD,x3),EqClass(RD,x1)];
  end;
  thus for c2,c3,c1 being Element of Class RD holds P[c2,c3,c1] from SchAux3(
  A2);
end;
