
theorem
  for V, V9, C, C9 being set st V c= V9 & C c= C9 holds the L_join of
SubstLatt (V, C) = (the L_join of SubstLatt (V9, C9))||the carrier of SubstLatt
  (V, C)
proof
  let V, V9, C, C9 be set;
  set K = SubstLatt (V, C), L = SubstLatt (V9, C9);
A1: SubstitutionSet (V, C) = the carrier of K by SUBSTLAT:def 4;
A2: dom (the L_join of L) = [:the carrier of L, the carrier of L:] by
FUNCT_2:def 1;
  reconsider B1 = the L_join of K as BinOp of the carrier of K;
  set B2 = (the L_join of L)||the carrier of K;
  assume
A3: V c= V9 & C c= C9;
  SubstitutionSet (V9, C9) = the carrier of L by SUBSTLAT:def 4;
  then the carrier of K c= the carrier of L by A1,A3,Th9;
  then
A4: dom B2 = [:the carrier of K, the carrier of K:] by A2,RELAT_1:62
,ZFMISC_1:96;
A5: SubstitutionSet (V, C) c= SubstitutionSet (V9, C9) by A3,Th9;
  rng B2 c= the carrier of K
  proof
    let x be object;
    assume x in rng B2;
    then consider y being object such that
A6: y in dom B2 and
A7: x = B2.y by FUNCT_1:def 3;
    consider y1, y2 being object such that
A8: y1 in the carrier of K & y2 in the carrier of K and
A9: y = [y1, y2] by A4,A6,ZFMISC_1:def 2;
    y1 in SubstitutionSet (V, C) & y2 in SubstitutionSet (V, C) by A8,
SUBSTLAT:def 4;
    then reconsider
    y19 = y1, y29 = y2 as Element of SubstitutionSet (V9, C9) by A5;
    reconsider y11 = y1, y22 = y2 as Element of SubstitutionSet (V, C) by A8,
SUBSTLAT:def 4;
    B2.y = (the L_join of L). (y1, y2) by A4,A6,A9,FUNCT_1:49
      .= mi (y19 \/ y29) by SUBSTLAT:def 4
      .= mi (y11 \/ y22) by A3,Th10;
    hence thesis by A1,A7;
  end;
  then reconsider B2 as BinOp of the carrier of K by A4,FUNCT_2:2;
  now
    let a, b be Element of K;
    reconsider a9 = a, b9 = b as Element of SubstitutionSet (V, C) by
SUBSTLAT:def 4;
    a9 in SubstitutionSet (V, C) & b9 in SubstitutionSet (V, C);
    then reconsider
    a1 = a, b1 = b as Element of SubstitutionSet (V9, C9) by A5;
    thus B1.(a, b) = mi (a9 \/ b9) by SUBSTLAT:def 4
      .= mi (a1 \/ b1) by A3,Th10
      .= (the L_join of L).(a, b) by SUBSTLAT:def 4
      .= B2.[a, b] by FUNCT_1:49
      .= B2.(a, b);
  end;
  hence thesis by BINOP_1:2;
end;
