reserve L,L1,L2 for Lattice,
  F1,F2 for Filter of L,
  p,q,r,s for Element of L,
  p1,q1,r1,s1 for Element of L1,
  p2,q2,r2,s2 for Element of L2,
  X,x,x1,x2,y,y1,y2 for set,
  D,D1,D2 for non empty set,
  R for Relation,
  RD for Equivalence_Relation of D,
  a,b,d for Element of D,
  a1,b1,c1 for Element of D1,
  a2,b2,c2 for Element of D2,
  B for B_Lattice,
  FB for Filter of B,
  I for I_Lattice,
  FI for Filter of I ,
  i,i1,i2,j,j1,j2,k for Element of I,
  f1,g1 for BinOp of D1,
  f2,g2 for BinOp of D2;
reserve F,G for BinOp of D,RD;

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;
