
theorem Th12:
  for A being non empty set for L being lower-bounded LATTICE st L
is modular for d being BiFunction of A,L st d is symmetric & d is u.t.i. for q
being Element of [:A,A,the carrier of L,the carrier of L:]
  st d.(q`1_4,q`2_4) <= (q`3_4)"\/"(q`4_4) holds new_bi_fun2(d,q) is u.t.i.
proof
  let A be non empty set;
  let L be lower-bounded LATTICE;
  assume
A1: L is modular;
  reconsider B = {{A}, {{A}}} as non empty set;
  let d be BiFunction of A,L;
  assume that
A2: d is symmetric and
A3: d is u.t.i.;
  let q be Element of [:A,A,the carrier of L,the carrier of L:];
  set x = q`1_4, y = q`2_4, a = q`3_4, b = q`4_4, f = new_bi_fun2(d,q);
  reconsider a,b as Element of L;
A4: for p,q,u being Element of new_set2 A st p in A & q in A & u in B holds
  f.(p,u) <= f.(p,q) "\/" f.(q,u)
  proof
    let p,q,u be Element of new_set2 A;
    assume
A5: p in A & q in A & u in B;
    per cases by A5,TARSKI:def 2;
    suppose
A6:   p in A & q in A & u = {A};
      then reconsider p9 = p, q9 = q as Element of A;
A7:   f.(p,q) = d.(p9,q9) by Def4;
      d.(p9,x) <= d.(p9,q9) "\/" d.(q9,x) by A3;
      then
A8:   d.(p9,x)"\/"a <= (d.(p9,q9) "\/" d.(q9,x))"\/"a by WAYBEL_1:2;
      f.(p,u) = d.(p9,x)"\/"a & f.(q,u) = d.(q9,x)"\/"a by A6,Def4;
      hence thesis by A7,A8,LATTICE3:14;
    end;
    suppose
A9:   p in A & q in A & u = {{A}};
      then reconsider p9 = p, q9 = q as Element of A;
A10:  f.(p,q) = d.(p9,q9) by Def4;
      d.(p9,y) <= d.(p9,q9) "\/" d.(q9,y) by A3;
      then
A11:  d.(p9,y)"\/"a <= (d.(p9,q9) "\/" d.(q9,y))"\/"a by WAYBEL_1:2;
      f.(p,u) = d.(p9,y)"\/"a & f.(q,u) = d.(q9,y)"\/"a by A9,Def4;
      hence thesis by A10,A11,LATTICE3:14;
    end;
  end;
  assume
A12: d.(q`1_4,q`2_4) <= (q`3_4)"\/"(q`4_4);
A13: for p,q,u being Element of new_set2 A st p in A & q in B & u in B holds
  f.(p,u) <= f.(p,q) "\/" f.(q,u)
  proof
    let p,q,u be Element of new_set2 A;
    assume
A14: p in A & q in B & u in B;
    per cases by A14,TARSKI:def 2;
    suppose
A15:  p in A & q = {A} & u = {A};
      then f.(p,q) "\/" f.(q,u) = Bottom L"\/"f.(p,q) by Def4
        .= f.(p,q) by WAYBEL_1:3;
      hence thesis by A15;
    end;
    suppose
A16:  p in A & q = {A} & u = {{A}};
      then reconsider p9 = p as Element of A;
      a <= a "\/" d.(x,y) by YELLOW_0:22;
      then
A17:  a"\/"((d.(x,y)"\/"a)"/\"b) = (a"\/"b)"/\"(a"\/" d.(x,y)) by A1;
      d.(p9,y) <= d.(p9,x)"\/"d.(x,y) by A3;
      then
A18:  d.(p9,y)"\/"a <= (d.(p9,x)"\/"d.(x,y))"\/"a by YELLOW_5:7;
      a <= a;
      then d.(x,y)"\/"a <= (a"\/"b)"\/"a by A12,YELLOW_3:3;
      then
      (d.(x,y)"\/"a)"/\"(d.(x,y)"\/"a) <= (d.(x,y)"\/"a)"/\"((a"\/"b)"\/"
      a) by YELLOW_5:6;
      then
A19:  d.(x,y)"\/"a = (d.(x,y)"\/"a)"/\"(d.(x,y)"\/"a) & d.(p9,x)"\/"((d.(
x,y)"\/"a )"/\"(d.(x,y)"\/"a)) <= d.(p9,x)"\/"((d.(x,y)"\/"a)"/\"((a"\/"b)"\/"
      a)) by WAYBEL_1:2,YELLOW_5:2;
      f.(p,q) = d.(p9,x)"\/"a & f.(q,u) = (d.(x,y)"\/"a)"/\"b by A16,Def4;
      then
      f.(p,q) "\/" f.(q,u) = d.(p9,x)"\/"((a"\/"b)"/\"(a"\/" d.(x,y))) by A17,
LATTICE3:14
        .= d.(p9,x)"\/"((d.(x,y)"\/"a)"/\"((a"\/"a)"\/"b)) by YELLOW_5:1
        .= d.(p9,x)"\/"((d.(x,y)"\/"a)"/\"((a"\/"b)"\/" a)) by LATTICE3:14;
      then (d.(p9,x)"\/"d.(x,y))"\/"a <= f.(p,q) "\/" f.(q,u) by A19,
LATTICE3:14;
      then d.(p9,y)"\/"a <= f.(p,q) "\/" f.(q,u) by A18,ORDERS_2:3;
      hence thesis by A16,Def4;
    end;
    suppose
A20:  p in A & q = {{A}} & u = {A};
      then reconsider p9 = p as Element of A;
      a <= a "\/" d.(x,y) by YELLOW_0:22;
      then
A21:  a"\/"((d.(x,y)"\/"a)"/\"b) = (a"\/"b)"/\"(a"\/" d.(x,y)) by A1;
      a <= a;
      then d.(x,y)"\/"a <= (a"\/"b)"\/"a by A12,YELLOW_3:3;
      then
      (d.(x,y)"\/"a)"/\"(d.(x,y)"\/"a) <= (d.(x,y)"\/"a)"/\"((a"\/"b)"\/"
      a) by YELLOW_5:6;
      then
A22:  d.(x,y)"\/"a = (d.(x,y)"\/"a)"/\"(d.(x,y)"\/"a) & d.(p9,y)"\/"((d.(
x,y)"\/"a )"/\"(d.(x,y)"\/"a)) <= d.(p9,y)"\/"((d.(x,y)"\/"a)"/\"((a"\/"b)"\/"
      a)) by WAYBEL_1:2,YELLOW_5:2;
      d.(y,x) = d.(x,y) by A2;
      then d.(p9,x) <= d.(p9,y)"\/"d.(x,y) by A3;
      then
A23:  d.(p9,x)"\/"a <= (d.(p9,y)"\/"d.(x,y))"\/"a by YELLOW_5:7;
      f.(p,q) = d.(p9,y)"\/"a & f.(q,u) = (d.(x,y)"\/"a)"/\"b by A20,Def4;
      then
      f.(p,q) "\/" f.(q,u) = d.(p9,y)"\/"((a"\/"b)"/\"(a"\/" d.(x,y))) by A21,
LATTICE3:14
        .= d.(p9,y)"\/"((d.(x,y)"\/"a)"/\"((a"\/"a)"\/"b)) by YELLOW_5:1
        .= d.(p9,y)"\/"((d.(x,y)"\/"a)"/\"((a"\/"b)"\/" a)) by LATTICE3:14;
      then (d.(p9,y)"\/"d.(x,y))"\/"a <= f.(p,q) "\/" f.(q,u) by A22,
LATTICE3:14;
      then d.(p9,x)"\/"a <= f.(p,q) "\/" f.(q,u) by A23,ORDERS_2:3;
      hence thesis by A20,Def4;
    end;
    suppose
A24:  p in A & q = {{A}} & u = {{A}};
      then f.(p,q) "\/" f.(q,u) = Bottom L"\/"f.(p,q) by Def4
        .= f.(p,q) by WAYBEL_1:3;
      hence thesis by A24;
    end;
  end;
A25: for p,q,u being Element of new_set2 A st p in B & q in A & u in B holds
  f.(p,u) <= f.(p,q) "\/" f.(q,u)
  proof
    let p,q,u be Element of new_set2 A;
    assume
A26: p in B & q in A & u in B;
    per cases by A26,TARSKI:def 2;
    suppose
      q in A & p = {A} & u = {A};
      then f.(p,u) = Bottom L by Def4;
      hence thesis by YELLOW_0:44;
    end;
    suppose
A27:  q in A & p = {A} & u = {{A}};
      then reconsider q9 = q as Element of A;
      d.(q9,x) = d.(x,q9) by A2;
      then
A28:  d.(x,y) <= d.(q9,x)"\/"d.(q9,y) by A3;
      f.(p,q) = d.(q9,x)"\/"a by A27,Def4;
      then f.(p,q) "\/" f.(q,u) = d.(q9,x)"\/"a"\/"(d.(q9,y)"\/"a) by A27,Def4
        .= d.(q9,x)"\/"(d.(q9,y)"\/"a)"\/"a by LATTICE3:14
        .= d.(q9,x)"\/"d.(q9,y)"\/"a"\/"a by LATTICE3:14
        .= d.(q9,x)"\/"d.(q9,y)"\/"(a"\/"a) by LATTICE3:14
        .= d.(q9,x)"\/"d.(q9,y)"\/"a by YELLOW_5:1;
      then
A29:  d.(x,y)"\/"a <= f.(p,q) "\/" f.(q,u) by A28,YELLOW_5:7;
A30:  (d.(x,y)"\/"a)"/\"b <= d.(x,y)"\/"a by YELLOW_0:23;
      f.(p,u) = (d.(x,y)"\/"a)"/\"b by A27,Def4;
      hence thesis by A29,A30,ORDERS_2:3;
    end;
    suppose
A31:  q in A & p = {{A}} & u = {A};
      then reconsider q9 = q as Element of A;
      d.(q9,x) = d.(x,q9) by A2;
      then
A32:  d.(x,y) <= d.(q9,x)"\/"d.(q9,y) by A3;
      f.(p,q) = d.(q9,y)"\/"a by A31,Def4;
      then
      f.(p,q) "\/" f.(q,u) = d.(q9,x)"\/"a"\/"(d.(q9,y)"\/"a) by A31,Def4
        .= d.(q9,x)"\/"(d.(q9,y)"\/"a)"\/"a by LATTICE3:14
        .= d.(q9,x)"\/"d.(q9,y)"\/"a"\/"a by LATTICE3:14
        .= d.(q9,x)"\/"d.(q9,y)"\/"(a"\/"a) by LATTICE3:14
        .= d.(q9,x)"\/"d.(q9,y)"\/"a by YELLOW_5:1;
      then
A33:  d.(x,y)"\/"a <= f.(p,q) "\/" f.(q,u) by A32,YELLOW_5:7;
A34:  (d.(x,y)"\/"a)"/\"b <= d.(x,y)"\/"a by YELLOW_0:23;
      f.(p,u) = (d.(x,y)"\/"a)"/\"b by A31,Def4;
      hence thesis by A33,A34,ORDERS_2:3;
    end;
    suppose
      q in A & p = {{A}} & u = {{A}};
      then f.(p,u) = Bottom L by Def4;
      hence thesis by YELLOW_0:44;
    end;
  end;
A35: for p,q,u being Element of new_set2 A st p in B & q in B & u in B
  holds f.(p,u) <= f.(p,q) "\/" f.(q,u)
  proof
    let p,q,u be Element of new_set2 A;
    assume
A36: p in B & q in B & u in B;
    per cases by A36,TARSKI:def 2;
    suppose
A37:  p = {A} & q = {A} & u = {A};
      Bottom L <= f.(p,q) "\/" f.(q,u) by YELLOW_0:44;
      hence thesis by A37,Def4;
    end;
    suppose
A38:  p = {A} & q = {A} & u = {{A}};
      then f.(p,q) "\/" f.(q,u) = Bottom L "\/" f.(p,u) by Def4
        .= Bottom L "\/" ((d.(x,y)"\/"a)"/\"b) by A38,Def4
        .= ((d.(x,y)"\/"a)"/\"b) by WAYBEL_1:3;
      hence thesis by A38,Def4;
    end;
    suppose
A39:  p = {A} & q = {{A}} & u = {A};
      Bottom L <= f.(p,q) "\/" f.(q,u) by YELLOW_0:44;
      hence thesis by A39,Def4;
    end;
    suppose
A40:  p = {A} & q = {{A}} & u = {{A}};
      then f.(p,q) "\/" f.(q,u) = ((d.(x,y)"\/"a)"/\"b)"\/"f.(q,u) by Def4
        .= Bottom L"\/"((d.(x,y)"\/"a)"/\"b) by A40,Def4
        .= ((d.(x,y)"\/"a)"/\"b) by WAYBEL_1:3;
      hence thesis by A40,Def4;
    end;
    suppose
A41:  p = {{A}} & q = {A} & u = {A};
      then f.(p,q) "\/" f.(q,u) = ((d.(x,y)"\/"a)"/\"b)"\/" f.(q,u) by Def4
        .= Bottom L"\/"((d.(x,y)"\/"a)"/\"b) by A41,Def4
        .= ((d.(x,y)"\/"a)"/\"b) by WAYBEL_1:3
        .= f.(p,q) by A41,Def4;
      hence thesis by A41;
    end;
    suppose
A42:  p = {{A}} & q = {A} & u = {{A}};
      Bottom L <= f.(p,q) "\/" f.(q,u) by YELLOW_0:44;
      hence thesis by A42,Def4;
    end;
    suppose
A43:  p = {{A}} & q = {{A}} & u = {A};
      then f.(p,q) "\/" f.(q,u) = Bottom L"\/" f.(p,u) by Def4
        .= Bottom L"\/"((d.(x,y)"\/"a)"/\"b) by A43,Def4
        .= ((d.(x,y)"\/"a)"/\"b) by WAYBEL_1:3;
      hence thesis by A43,Def4;
    end;
    suppose
A44:  p = {{A}} & q = {{A}} & u = {{A}};
      Bottom L <= f.(p,q) "\/" f.(q,u) by YELLOW_0:44;
      hence thesis by A44,Def4;
    end;
  end;
A45: for p,q,u being Element of new_set2 A st p in B & q in B & u in A
  holds f.(p,u) <= f.(p,q) "\/" f.(q,u)
  proof
    let p,q,u be Element of new_set2 A;
    assume
A46: p in B & q in B & u in A;
    per cases by A46,TARSKI:def 2;
    suppose
A47:  u in A & q = {A} & p = {A};
      then f.(p,q)"\/"f.(q,u) = Bottom L"\/"f.(q,u) by Def4
        .= f.(p,u) by A47,WAYBEL_1:3;
      hence thesis;
    end;
    suppose
A48:  u in A & q = {A} & p = {{A}};
      then reconsider u9 = u as Element of A;
      a <= a "\/" d.(x,y) by YELLOW_0:22;
      then
A49:  a"\/"((d.(x,y)"\/"a)"/\"b) = (a"\/"b)"/\"(a"\/" d.(x,y)) by A1;
      d.(u9,y) <= d.(u9,x)"\/"d.(x,y) by A3;
      then
A50:  d.(u9,y)"\/"a <= (d.(u9,x)"\/"d.(x,y))"\/"a by YELLOW_5:7;
      a <= a;
      then d.(x,y)"\/"a <= (a"\/"b)"\/"a by A12,YELLOW_3:3;
      then (d.(x,y)"\/"a)"/\"(d.(x,y)"\/"a) <= (d.(x,y)"\/"a)"/\"((a"\/"b)
      "\/"a) by YELLOW_5:6;
      then
A51:  d.(x,y)"\/"a = (d.(x,y)"\/"a)"/\"(d.(x,y)"\/"a) & d.(u9,x)"\/"((d.
(x,y)"\/"a )"/\"(d.(x,y)"\/"a)) <= d.(u9,x)"\/"((d.(x,y)"\/"a)"/\"((a"\/"b)"\/"
      a)) by WAYBEL_1:2,YELLOW_5:2;
      f.(p,q) = (d.(x,y)"\/"a)"/\"b & f.(q,u) = d.(u9,x)"\/"a by A48,Def4;
      then f.(p,q) "\/" f.(q,u) = d.(u9,x)"\/"((a"\/"b)"/\"(a"\/" d.(x,y)))
      by A49,LATTICE3:14
        .= d.(u9,x)"\/"((d.(x,y)"\/"a)"/\"((a"\/"a)"\/"b)) by YELLOW_5:1
        .= d.(u9,x)"\/"((d.(x,y)"\/"a)"/\"((a"\/"b)"\/" a)) by LATTICE3:14;
      then (d.(u9,x)"\/"d.(x,y))"\/"a <= f.(p,q) "\/" f.(q,u) by A51,
LATTICE3:14;
      then d.(u9,y)"\/"a <= f.(p,q) "\/" f.(q,u) by A50,ORDERS_2:3;
      hence thesis by A48,Def4;
    end;
    suppose
A52:  u in A & q = {{A}} & p = {A};
      then reconsider u9 = u as Element of A;
      a <= a "\/" d.(x,y) by YELLOW_0:22;
      then
A53:  a"\/"((d.(x,y)"\/"a)"/\"b) = (a"\/"b)"/\"(a"\/" d.(x,y)) by A1;
      a <= a;
      then d.(x,y)"\/"a <= (a"\/"b)"\/"a by A12,YELLOW_3:3;
      then (d.(x,y)"\/"a)"/\"(d.(x,y)"\/"a) <= (d.(x,y)"\/"a)"/\"((a"\/"b)
      "\/"a) by YELLOW_5:6;
      then
A54:  d.(x,y)"\/"a = (d.(x,y)"\/"a)"/\"(d.(x,y)"\/"a) & d.(u9,y)"\/"((d.
(x,y)"\/"a )"/\"(d.(x,y)"\/"a)) <= d.(u9,y)"\/"((d.(x,y)"\/"a)"/\"((a"\/"b)"\/"
      a)) by WAYBEL_1:2,YELLOW_5:2;
      d.(y,x) = d.(x,y) by A2;
      then d.(u9,x) <= d.(u9,y)"\/"d.(x,y) by A3;
      then
A55:  d.(u9,x)"\/"a <= (d.(u9,y)"\/"d.(x,y))"\/"a by YELLOW_5:7;
      f.(p,q) = (d.(x,y)"\/"a)"/\"b & f.(q,u) = d.(u9,y)"\/"a by A52,Def4;
      then f.(p,q) "\/" f.(q,u) = d.(u9,y)"\/"((a"\/"b)"/\"(a"\/" d.(x,y)))
      by A53,LATTICE3:14
        .= d.(u9,y)"\/"((d.(x,y)"\/"a)"/\"((a"\/"a)"\/"b)) by YELLOW_5:1
        .= d.(u9,y)"\/"((d.(x,y)"\/"a)"/\"((a"\/"b)"\/" a)) by LATTICE3:14;
      then (d.(u9,y)"\/"d.(x,y))"\/"a <= f.(p,q) "\/" f.(q,u) by A54,
LATTICE3:14;
      then d.(u9,x)"\/"a <= f.(p,q) "\/" f.(q,u) by A55,ORDERS_2:3;
      hence thesis by A52,Def4;
    end;
    suppose
A56:  u in A & q = {{A}} & p = {{A}};
      then f.(p,q)"\/"f.(q,u) = Bottom L"\/"f.(q,u) by Def4
        .= f.(p,u) by A56,WAYBEL_1:3;
      hence thesis;
    end;
  end;
A57: for p,q,u being Element of new_set2 A st p in B & q in A & u in A holds
  f.(p,u) <= f.(p,q) "\/" f.(q,u)
  proof
    let p,q,u be Element of new_set2 A;
    assume that
A58: p in B and
A59: q in A & u in A;
    reconsider q9 = q, u9 = u as Element of A by A59;
    per cases by A58,A59,TARSKI:def 2;
    suppose
A60:  p = {A} & q in A & u in A;
      d.(u9,x) <= d.(u9,q9) "\/" d.(q9,x) by A3;
      then d.(u9,x) <= d.(q9,u9) "\/" d.(q9,x) by A2;
      then d.(u9,x)"\/"a <= (d.(q9,x)"\/"d.(q9,u9))"\/"a by WAYBEL_1:2;
      then
A61:  d.(u9,x)"\/"a <= (d.(q9,x)"\/"a)"\/"d.(q9,u9) by LATTICE3:14;
A62:  f.(q,u) = d.(q9,u9) by Def4;
      f.(p,q) = d.(q9,x)"\/"a by A60,Def4;
      hence thesis by A60,A62,A61,Def4;
    end;
    suppose
A63:  p = {{A}} & q in A & u in A;
      d.(u9,y) <= d.(u9,q9) "\/" d.(q9,y) by A3;
      then d.(u9,y) <= d.(q9,u9) "\/" d.(q9,y) by A2;
      then d.(u9,y)"\/"a <= (d.(q9,y)"\/"d.(q9,u9))"\/"a by WAYBEL_1:2;
      then
A64:  d.(u9,y)"\/"a <= (d.(q9,y)"\/"a)"\/"d.(q9,u9) by LATTICE3:14;
A65:  f.(q,u) = d.(q9,u9) by Def4;
      f.(p,q) = d.(q9,y)"\/"a by A63,Def4;
      hence thesis by A63,A65,A64,Def4;
    end;
  end;
A66: for p,q,u being Element of new_set2 A st p in A & q in B & u in A holds
  f.(p,u) <= f.(p,q) "\/" f.(q,u)
  proof
    let p,q,u be Element of new_set2 A;
    assume
A67: p in A & q in B & u in A;
    per cases by A67,TARSKI:def 2;
    suppose
A68:  p in A & u in A & q = {A};
      then reconsider p9 = p, u9 = u as Element of A;
      d.(p9,u9) <= d.(p9,x) "\/" d.(x,u9) by A3;
      then
A69:  d.(p9,x) "\/" d.(u9,x) <= (d.(p9,x) "\/" d.(u9,x))"\/"a & d.(p9,u9)
      <= d.(p9,x) "\/" d.(u9,x) by A2,YELLOW_0:22;
      (d.(p9,x)"\/"d.(u9,x))"\/"a = d.(p9,x)"\/"(d.(u9,x)"\/"a) by LATTICE3:14
        .= d.(p9,x)"\/"(d.(u9,x)"\/"(a"\/"a)) by YELLOW_5:1
        .= d.(p9,x)"\/"((d.(u9,x)"\/"a)"\/"a) by LATTICE3:14
        .= (d.(p9,x)"\/"a) "\/" (d.(u9,x)"\/"a) by LATTICE3:14;
      then
A70:  d.(p9,u9) <= (d.(p9,x)"\/"a) "\/" (d.(u9,x)"\/"a) by A69,ORDERS_2:3;
      f.(p,q) = d.(p9,x)"\/"a & f.(q,u) = d.(u9,x)"\/"a by A68,Def4;
      hence thesis by A70,Def4;
    end;
    suppose
A71:  p in A & u in A & q = {{A}};
      then reconsider p9 = p, u9 = u as Element of A;
      d.(p9,u9) <= d.(p9,y) "\/" d.(y,u9) by A3;
      then
A72:  d.(p9,y) "\/" d.(u9,y) <= (d.(p9,y) "\/" d.(u9,y))"\/"a & d.(p9,u9)
      <= d.(p9,y) "\/" d.(u9,y) by A2,YELLOW_0:22;
      (d.(p9,y)"\/"d.(u9,y))"\/"a = d.(p9,y)"\/"(d.(u9,y)"\/"a) by LATTICE3:14
        .= d.(p9,y)"\/"(d.(u9,y)"\/"(a"\/"a)) by YELLOW_5:1
        .= d.(p9,y)"\/"((d.(u9,y)"\/"a)"\/"a) by LATTICE3:14
        .= (d.(p9,y)"\/"a)"\/"(d.(u9,y)"\/"a) by LATTICE3:14;
      then
A73:  d.(p9,u9) <= (d.(p9,y)"\/"a) "\/" (d.(u9,y)"\/"a) by A72,ORDERS_2:3;
      f.(p,q) = d.(p9,y)"\/"a & f.(q,u) = d.(u9,y)"\/"a by A71,Def4;
      hence thesis by A73,Def4;
    end;
  end;
A74: for p,q,u being Element of new_set2 A st p in A & q in A & u in A holds
  f.(p,u) <= f.(p,q) "\/" f.(q,u)
  proof
    let p,q,u be Element of new_set2 A;
    assume p in A & q in A & u in A;
    then reconsider p9 = p, q9 = q, u9 = u as Element of A;
A75: f.(q,u) = d.(q9,u9) by Def4;
    f.(p,u) = d.(p9,u9) & f.(p,q) = d.(p9,q9) by Def4;
    hence thesis by A3,A75;
  end;
  for p,q,u being Element of new_set2 A holds f.(p,u) <= f.(p,q) "\/" f.
  ( q, u )
  proof
    let p,q,u be Element of new_set2 A;
    per cases by XBOOLE_0:def 3;
    suppose
      p in A & q in A & u in A;
      hence thesis by A74;
    end;
    suppose
      p in A & q in A & u in B;
      hence thesis by A4;
    end;
    suppose
      p in A & q in B & u in A;
      hence thesis by A66;
    end;
    suppose
      p in A & q in B & u in B;
      hence thesis by A13;
    end;
    suppose
      p in B & q in A & u in A;
      hence thesis by A57;
    end;
    suppose
      p in B & q in A & u in B;
      hence thesis by A25;
    end;
    suppose
      p in B & q in B & u in A;
      hence thesis by A45;
    end;
    suppose
      p in B & q in B & u in B;
      hence thesis by A35;
    end;
  end;
  hence thesis;
end;
