reserve X for non empty set;
reserve e,e1,e2,e19,e29 for Equivalence_Relation of X,
  x,y,x9,y9 for set;
reserve A for non empty set,
  L for lower-bounded LATTICE;

theorem Th18:
  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_fun(d,q) is u.t.i.
proof
  let d be BiFunction of A,L;
  assume that
A1: d is symmetric and
A2: d is u.t.i.;
  reconsider B = {{A}, {{A}}, {{{A}}}} as non empty set;
  let q be Element of [:A,A,the carrier of L,the carrier of L:];
  set x = q`1_4, y = q`2_4, f = new_bi_fun(d,q);
  reconsider a = q`3_4,b = q`4_4 as Element of L;
A3: for p,q,u being Element of new_set 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_set A;
    assume
A4: p in A & q in B & u in A;
    per cases by A4,ENUMSET1:def 1;
    suppose
A5:   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 A2;
      then
A6:   d.(p9,x) "\/" d.(u9,x) <= (d.(p9,x) "\/" d.(u9,x))"\/"a & d.(p9,u9)
      <= d.(p9,x) "\/" d.(u9,x) by A1,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
A7:   d.(p9,u9) <= (d.(p9,x)"\/"a) "\/" (d.(u9,x)"\/"a) by A6,ORDERS_2:3;
      f.(p,q) = d.(p9,x)"\/"a & f.(q,u) = d.(u9,x)"\/"a by A5,Def10;
      hence thesis by A7,Def10;
    end;
    suppose
A8:   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 A2;
      then
A9:   d.(p9,x) "\/" d.(u9,x) <= (d.(p9,x) "\/" d.(u9,x))"\/"(a"\/" b) & d
      .(p9,u9) <= d.(p9,x) "\/" d.(u9,x) by A1,YELLOW_0:22;
      (d.(p9,x)"\/"d.(u9,x))"\/"(a"\/"b) = d.(p9,x)"\/"(d.(u9,x)"\/"(a
      "\/" b)) by LATTICE3:14
        .= d.(p9,x)"\/"(d.(u9,x)"\/"((a"\/"b)"\/"(a"\/"b))) by YELLOW_5:1
        .= d.(p9,x)"\/"((d.(u9,x)"\/"(a"\/"b))"\/"(a"\/"b)) by LATTICE3:14
        .= (d.(p9,x)"\/"(a"\/"b))"\/"(d.(u9,x)"\/"(a"\/"b)) by LATTICE3:14
        .= (d.(p9,x)"\/"a"\/"b) "\/" (d.(u9,x)"\/"(a"\/"b)) by LATTICE3:14
        .= (d.(p9,x)"\/"a"\/"b) "\/" (d.(u9,x)"\/"a"\/"b) by LATTICE3:14;
      then
A10:  d.(p9,u9) <= (d.(p9,x)"\/"a"\/"b) "\/" (d.(u9,x)"\/"a"\/" b) by A9,
ORDERS_2:3;
      f.(p,q) = d.(p9,x)"\/"a"\/"b & f.(q,u) = d.(u9,x)"\/"a"\/"b by A8,Def10;
      hence thesis by A10,Def10;
    end;
    suppose
A11:  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 A2;
      then
A12:  d.(p9,y) "\/" d.(u9,y) <= (d.(p9,y) "\/" d.(u9,y))"\/"b & d.(p9,u9)
      <= d.(p9,y) "\/" d.(u9,y) by A1,YELLOW_0:22;
      (d.(p9,y)"\/"d.(u9,y))"\/"b = d.(p9,y)"\/"(d.(u9,y)"\/"b) by LATTICE3:14
        .= d.(p9,y)"\/"(d.(u9,y)"\/"(b"\/"b)) by YELLOW_5:1
        .= d.(p9,y)"\/"((d.(u9,y)"\/"b)"\/"b) by LATTICE3:14
        .= (d.(p9,y)"\/"b) "\/" (d.(u9,y)"\/"b) by LATTICE3:14;
      then
A13:  d.(p9,u9) <= (d.(p9,y)"\/"b) "\/" (d.(u9,y)"\/"b) by A12,ORDERS_2:3;
      f.(p,q) = d.(p9,y)"\/"b & f.(q,u) = d.(u9,y)"\/"b by A11,Def10;
      hence thesis by A13,Def10;
    end;
  end;
  assume
A14: d.(q`1_4,q`2_4) <= (q`3_4)"\/"(q`4_4);
A15: for p,q,u being Element of new_set 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_set A;
    assume that
A16: p in B & q in B and
A17: u in A;
    reconsider u9 = u as Element of A by A17;
    per cases by A16,A17,ENUMSET1:def 1;
    suppose
A18:  u in A & q = {A} & p = {A};
      then f.(p,q)"\/"f.(q,u) = Bottom L"\/" f.(q,u) by Def10
        .= f.(p,u) by A18,WAYBEL_1:3;
      hence thesis;
    end;
    suppose
A19:  u in A & q = {A} & p = {{A}};
      then f.(p,q)"\/"f.(q,u) = b"\/"f.(q,u) by Def10
        .= d.(u9,x)"\/"a"\/" b by A19,Def10;
      hence thesis by A19,Def10;
    end;
    suppose
A20:  u in A & q = {A} & p = {{{A}}};
      b"\/"(a"\/"b) = (b"\/"b)"\/"a by LATTICE3:14
        .= b"\/"a by YELLOW_5:1
        .= b"\/"(a"\/"a) by YELLOW_5:1
        .= a"\/"(a"\/"b) by LATTICE3:14;
      then
A21:  (d.(u9,x)"\/"b)"\/"(a"\/"b) = d.(u9,x)"\/"(a"\/"(a"\/" b)) by LATTICE3:14
        .= (a"\/"b)"\/"(d.(u9,x)"\/"a) by LATTICE3:14
        .= f.(p,q)"\/"(d.(u9,x)"\/" a) by A20,Def10;
      d.(u9,y) <= d.(u9,x)"\/"d.(x,y) by A2;
      then
A22:  d.(u9,y)"\/"b <= (d.(u9,x)"\/"d.(x,y))"\/"b by WAYBEL_1:2;
      d.(u9,x)"\/"b <= d.(u9,x)"\/"b;
      then
A23:  (d.(u9,x)"\/"d.(x,y))"\/"b = (d.(u9,x)"\/"b)"\/"d.(x,y) & (d.(u9,x
      )"\/"b) "\/"d.(x,y) <= (d.(u9,x)"\/"b)"\/"(a"\/" b) by A14,LATTICE3:14
,YELLOW_3:3;
      f.(p,u) = d.(u9,y)"\/"b by A20,Def10;
      then f.(p,u) <= (d.(u9,x)"\/"b)"\/"(a"\/"b) by A22,A23,ORDERS_2:3;
      hence thesis by A20,A21,Def10;
    end;
    suppose
A24:  u in A & q = {{A}} & p = {A};
      then f.(p,q)"\/"f.(q,u) = b"\/"f.(q,u) by Def10
        .= b"\/"(d.(u9,x)"\/"a"\/" b) by A24,Def10
        .= b"\/"(b"\/"f.(p,u)) by A24,Def10
        .= (b"\/"b)"\/"f.(p,u) by LATTICE3:14
        .= b"\/"f.(p,u) by YELLOW_5:1;
      hence thesis by YELLOW_0:22;
    end;
    suppose
A25:  u in A & q = {{A}} & p = {{A}};
      then f.(p,q)"\/"f.(q,u) = Bottom L"\/" f.(q,u) by Def10
        .= f.(p,u) by A25,WAYBEL_1:3;
      hence thesis;
    end;
    suppose
A26:  u in A & q = {{A}} & p = {{{A}}};
      b"\/"(a"\/"b) = (b"\/"b)"\/"a by LATTICE3:14
        .= b"\/"a by YELLOW_5:1
        .= b"\/"(a"\/"a) by YELLOW_5:1
        .= (a"\/"b)"\/"a by LATTICE3:14;
      then
A27:  (d.(u9,x)"\/"b)"\/"(a"\/"b) = d.(u9,x)"\/"((a"\/"b)"\/"a) by LATTICE3:14
        .= (d.(u9,x)"\/"(a"\/"b))"\/"a by LATTICE3:14
        .= (d.(u9,x)"\/"a"\/"b)"\/"a by LATTICE3:14
        .= f.(p,q)"\/"(d.(u9,x)"\/"a"\/"b) by A26,Def10;
      d.(u9,x)"\/"b <= d.(u9,x)"\/"b;
      then
A28:  (d.(u9,x)"\/"d.(x,y))"\/"b = (d.(u9,x)"\/"b)"\/"d.(x,y) & (d.(u9,x
      )"\/"b) "\/"d.(x,y) <= (d.(u9,x)"\/"b)"\/"(a"\/" b) by A14,LATTICE3:14
,YELLOW_3:3;
      d.(u9,y) <= d.(u9,x)"\/"d.(x,y) by A2;
      then
A29:  d.(u9,y)"\/"b <= (d.(u9,x)"\/"d.(x,y))"\/"b by WAYBEL_1:2;
      f.(p,u) = d.(u9,y)"\/"b by A26,Def10;
      then f.(p,u) <= (d.(u9,x)"\/"b)"\/"(a"\/"b) by A29,A28,ORDERS_2:3;
      hence thesis by A26,A27,Def10;
    end;
    suppose
A30:  u in A & q = {{{A}}} & p = {A};
A31:  a"\/"(a"\/"b) = (a"\/"a)"\/"b by LATTICE3:14
        .= a"\/"b by YELLOW_5:1
        .= a"\/"(b"\/"b) by YELLOW_5:1
        .= b"\/"(a"\/"b) by LATTICE3:14;
A32:  (a"\/"d.(u9,y))"\/"(a"\/"b) = d.(u9,y)"\/"(a"\/"(a"\/"b)) by LATTICE3:14
        .= (d.(u9,y)"\/"b)"\/"(a"\/"b) by A31,LATTICE3:14
        .= f.(p,q)"\/"(d.(u9,y)"\/"b) by A30,Def10;
      a"\/"d.(u9,y) <= a"\/"d.(u9,y);
      then
A33:  (a"\/"d.(u9,y))"\/"d.(x,y) <= (a"\/"d.(u9,y))"\/"(a"\/" b) by A14,
YELLOW_3:3;
      d.(u9,x) <= d.(u9,y)"\/"d.(y,x) by A2;
      then
A34:  d.(u9,x)"\/"a <= (d.(u9,y)"\/"d.(y,x))"\/"a by WAYBEL_1:2;
A35:  (d.(u9,y)"\/"d.(y,x))"\/"a = d.(y,x)"\/"(d.(u9,y)"\/"a) by LATTICE3:14
        .= (a"\/"d.(u9,y))"\/"d.(x,y) by A1;
      f.(p,u) = d.(u9,x)"\/"a by A30,Def10;
      then f.(p,u) <= (a"\/"d.(u9,y))"\/"(a"\/"b) by A34,A35,A33,ORDERS_2:3;
      hence thesis by A30,A32,Def10;
    end;
    suppose
A36:  u in A & q = {{{A}}} & p = {{A}};
      then
A37:  f.(p,u) = d.(u9,x)"\/"a"\/"b by Def10
        .= d.(u9,x)"\/"(a"\/" b) by LATTICE3:14;
      (a"\/"b)"\/"d.(u9,y) <= (a"\/"b)"\/"d.(u9,y);
      then
A38:  ((a"\/"b)"\/"d.(u9,y))"\/"d.(x,y) <= ((a"\/"b)"\/"d.(u9,y))"\/"(a
      "\/"b) by A14,YELLOW_3:3;
      d.(u9,x) <= d.(u9,y)"\/"d.(y,x) by A2;
      then
A39:  d.(u9,x)"\/"(a"\/"b) <= (d.(u9,y)"\/"d.(y,x))"\/"(a"\/"b) by WAYBEL_1:2;
A40:  (d.(u9,y)"\/"d.(y,x))"\/"(a"\/"b) = ((a"\/"b)"\/"d.(u9,y))"\/" d.(
      y,x) by LATTICE3:14
        .= ((a"\/"b)"\/"d.(u9,y))"\/"d.(x,y) by A1;
      f.(p,q)"\/"f.(q,u) = a"\/"f.(q,u) by A36,Def10
        .= a"\/"(b"\/" d.(u9,y)) by A36,Def10
        .= (a"\/"b)"\/"d.(u9,y) by LATTICE3:14
        .= ((a"\/"b)"\/"(a"\/"b))"\/"d.(u9,y) by YELLOW_5:1
        .= (a"\/"b)"\/"(d.(u9,y)"\/"(a"\/"b)) by LATTICE3:14;
      hence thesis by A37,A39,A40,A38,ORDERS_2:3;
    end;
    suppose
A41:  u in A & q = {{{A}}} & p = {{{A}}};
      then f.(p,q)"\/"f.(q,u) = Bottom L"\/" f.(q,u) by Def10
        .= f.(p,u) by A41,WAYBEL_1:3;
      hence thesis;
    end;
  end;
A42: for p,q,u being Element of new_set 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_set A;
    assume that
A43: p in B and
A44: q in A & u in A;
    reconsider q9 = q, u9 = u as Element of A by A44;
    per cases by A43,A44,ENUMSET1:def 1;
    suppose
A45:  p = {A} & q in A & u in A;
      d.(u9,x) <= d.(u9,q9) "\/" d.(q9,x) by A2;
      then d.(u9,x) <= d.(q9,u9) "\/" d.(q9,x) by A1;
      then d.(u9,x)"\/"a <= (d.(q9,x)"\/"d.(q9,u9))"\/"a by WAYBEL_1:2;
      then
A46:  d.(u9,x)"\/"a <= (d.(q9,x)"\/"a)"\/"d.(q9,u9) by LATTICE3:14;
A47:  f.(q,u) = d.(q9,u9) by Def10;
      f.(p,q) = d.(q9,x)"\/"a by A45,Def10;
      hence thesis by A45,A47,A46,Def10;
    end;
    suppose
A48:  p = {{A}} & q in A & u in A;
      d.(u9,x) <= d.(u9,q9) "\/" d.(q9,x) by A2;
      then d.(u9,x) <= d.(q9,u9) "\/" d.(q9,x) by A1;
      then d.(u9,x)"\/"(a"\/"b) <= (d.(q9,x)"\/"d.(q9,u9))"\/"(a"\/" b) by
WAYBEL_1:2;
      then
d.(u9,x)"\/"a"\/"b <= (d.(q9,x)"\/"d.(q9,u9))"\/"(a"\/" b) by LATTICE3:14;
      then
d.(u9,x)"\/"a"\/"b <= (d.(q9,x)"\/"(a"\/"b))"\/" d.(q9,u9) by LATTICE3:14;
      then
A49:  d.(u9,x)"\/"a"\/"b <= (d.(q9,x)"\/"a"\/"b)"\/"d.(q9,u9) by LATTICE3:14;
A50:  f.(q,u) = d.(q9,u9) by Def10;
      f.(p,q) = d.(q9,x)"\/"a"\/"b by A48,Def10;
      hence thesis by A48,A50,A49,Def10;
    end;
    suppose
A51:  p = {{{A}}} & q in A & u in A;
      d.(u9,y) <= d.(u9,q9) "\/" d.(q9,y) by A2;
      then d.(u9,y) <= d.(q9,u9) "\/" d.(q9,y) by A1;
      then d.(u9,y)"\/"b <= (d.(q9,y)"\/"d.(q9,u9))"\/"b by WAYBEL_1:2;
      then
A52:  d.(u9,y)"\/"b <= (d.(q9,y)"\/"b)"\/"d.(q9,u9) by LATTICE3:14;
A53:  f.(q,u) = d.(q9,u9) by Def10;
      f.(p,q) = d.(q9,y)"\/"b by A51,Def10;
      hence thesis by A51,A53,A52,Def10;
    end;
  end;
A54: for p,q,u being Element of new_set 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_set A;
    assume
A55: p in A & q in A & u in B;
    per cases by A55,ENUMSET1:def 1;
    suppose
A56:  p in A & q in A & u = {A};
      then reconsider p9 = p, q9 = q as Element of A;
A57:  f.(p,q) = d.(p9,q9) by Def10;
      d.(p9,x) <= d.(p9,q9) "\/" d.(q9,x) by A2;
      then
A58:  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 A56,Def10;
      hence thesis by A57,A58,LATTICE3:14;
    end;
    suppose
A59:  p in A & q in A & u = {{A}};
      then reconsider p9 = p, q9 = q as Element of A;
A60:  f.(p,q) = d.(p9,q9) by Def10;
      d.(p9,x) <= d.(p9,q9) "\/" d.(q9,x) by A2;
      then d.(p9,x)"\/"a <= (d.(p9,q9) "\/" d.(q9,x))"\/"a by WAYBEL_1:2;
      then (d.(p9,x)"\/"a)"\/"b <= ((d.(p9,q9) "\/" d.(q9,x))"\/"a)"\/" b by
WAYBEL_1:2;
      then
A61:  d.(p9,x)"\/"a"\/"b <= (d.(p9,q9) "\/" (d.(q9,x)"\/"a))"\/"b by
LATTICE3:14;
      f.(p,u) = d.(p9,x)"\/"a"\/"b & f.(q,u) = d.(q9,x)"\/"a"\/"b by A59,Def10;
      hence thesis by A60,A61,LATTICE3:14;
    end;
    suppose
A62:  p in A & q in A & u = {{{A}}};
      then reconsider p9 = p, q9 = q as Element of A;
A63:  f.(p,q) = d.(p9,q9) by Def10;
      d.(p9,y) <= d.(p9,q9) "\/" d.(q9,y) by A2;
      then
A64:  d.(p9,y)"\/"b <= (d.(p9,q9) "\/" d.(q9,y))"\/"b by WAYBEL_1:2;
      f.(p,u) = d.(p9,y)"\/"b & f.(q,u) = d.(q9,y)"\/"b by A62,Def10;
      hence thesis by A63,A64,LATTICE3:14;
    end;
  end;
A65: for p,q,u being Element of new_set 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_set A;
    assume
A66: p in B & q in B & u in B;
    per cases by A66,ENUMSET1:def 1;
    suppose
A67:  p = {A} & q = {A} & u = {A};
      Bottom L <= f.(p,q) "\/" f.(q,u) by YELLOW_0:44;
      hence thesis by A67,Def10;
    end;
    suppose
A68:  p = {A} & q = {A} & u = {{A}};
      then f.(p,q) "\/" f.(q,u) = Bottom L"\/"f.(p,u) by Def10
        .= Bottom L"\/"b by A68,Def10
        .= b by WAYBEL_1:3;
      hence thesis by A68,Def10;
    end;
    suppose
A69:  p = {A} & q = {A} & u = {{{A}}};
      then f.(p,q) "\/" f.(q,u) = Bottom L"\/"f.(p,u) by Def10
        .= Bottom L"\/"(a"\/"b) by A69,Def10
        .= a"\/"b by WAYBEL_1:3;
      hence thesis by A69,Def10;
    end;
    suppose
A70:  p = {A} & q = {{A}} & u = {A};
      Bottom L <= f.(p,q) "\/" f.(q,u) by YELLOW_0:44;
      hence thesis by A70,Def10;
    end;
    suppose
A71:  p = {A} & q = {{A}} & u = {{A}};
      then f.(p,q) "\/" f.(q,u) = b"\/"f.(q,u) by Def10
        .= Bottom L"\/"b by A71,Def10
        .= b by WAYBEL_1:3;
      hence thesis by A71,Def10;
    end;
    suppose
A72:  p = {A} & q = {{A}} & u = {{{A}}};
      then f.(p,q) "\/" f.(q,u) = b"\/"f.(q,u) by Def10
        .= a"\/"b by A72,Def10;
      hence thesis by A72,Def10;
    end;
    suppose
A73:  p = {A} & q = {{{A}}} & u = {A};
      Bottom L <= f.(p,q) "\/" f.(q,u) by YELLOW_0:44;
      hence thesis by A73,Def10;
    end;
    suppose
A74:  p = {A} & q = {{{A}}} & u = {{A}};
      then
A75:  f.(p,u) = b by Def10;
      f.(p,q) "\/" f.(q,u) = (a"\/"b)"\/" f.(q,u) by A74,Def10
        .= (b"\/"a) "\/" a by A74,Def10
        .= b"\/"(a"\/"a) by LATTICE3:14
        .= b"\/"a by YELLOW_5:1;
      hence thesis by A75,YELLOW_0:22;
    end;
    suppose
A76:  p = {A} & q = {{{A}}} & u = {{{A}}};
      then f.(p,q) "\/" f.(q,u) = (a"\/"b)"\/" f.(q,u) by Def10
        .= Bottom L"\/"(a "\/" b) by A76,Def10
        .= a"\/"b by WAYBEL_1:3
        .= f.(p,q) by A76,Def10;
      hence thesis by A76;
    end;
    suppose
A77:  p = {{A}} & q = {A} & u = {A};
      then f.(p,q) "\/" f.(q,u) = b"\/" f.(q,u) by Def10
        .= Bottom L"\/"b by A77,Def10
        .= b by WAYBEL_1:3
        .= f.(p,q) by A77,Def10;
      hence thesis by A77;
    end;
    suppose
A78:  p = {{A}} & q = {A} & u = {{A}};
      Bottom L <= f.(p,q) "\/" f.(q,u) by YELLOW_0:44;
      hence thesis by A78,Def10;
    end;
    suppose
A79:  p = {{A}} & q = {A} & u = {{{A}}};
      then
A80:  f.(p,u) = a by Def10;
      f.(p,q) "\/" f.(q,u) = b"\/" f.(q,u) by A79,Def10
        .= b"\/"(b"\/" a) by A79,Def10
        .= (b"\/"b)"\/"a by LATTICE3:14
        .= b"\/"a by YELLOW_5:1;
      hence thesis by A80,YELLOW_0:22;
    end;
    suppose
A81:  p = {{A}} & q = {{A}} & u = {A};
      then f.(p,q) "\/" f.(q,u) = Bottom L"\/" f.(p,u) by Def10
        .= Bottom L"\/"b by A81,Def10
        .= b by WAYBEL_1:3;
      hence thesis by A81,Def10;
    end;
    suppose
A82:  p = {{A}} & q = {{A}} & u = {{A}};
      Bottom L <= f.(p,q) "\/" f.(q,u) by YELLOW_0:44;
      hence thesis by A82,Def10;
    end;
    suppose
A83:  p = {{A}} & q = {{A}} & u = {{{A}}};
      then f.(p,q) "\/" f.(q,u) = Bottom L"\/" f.(p,u) by Def10
        .= Bottom L"\/"a by A83,Def10
        .= a by WAYBEL_1:3;
      hence thesis by A83,Def10;
    end;
    suppose
A84:  p = {{A}} & q = {{{A}}} & u = {A};
      then
A85:  f.(p,u) = b by Def10;
      f.(p,q) "\/" f.(q,u) = a"\/" f.(q,u) by A84,Def10
        .= a"\/"(a"\/" b) by A84,Def10
        .= (a"\/"a)"\/"b by LATTICE3:14
        .= a"\/"b by YELLOW_5:1;
      hence thesis by A85,YELLOW_0:22;
    end;
    suppose
A86:  p = {{A}} & q = {{{A}}} & u = {{A}};
      Bottom L <= f.(p,q) "\/" f.(q,u) by YELLOW_0:44;
      hence thesis by A86,Def10;
    end;
    suppose
A87:  p = {{A}} & q = {{{A}}} & u = {{{A}}};
      then f.(p,q) "\/" f.(q,u) = a"\/" f.(q,u) by Def10
        .= Bottom L"\/"a by A87,Def10
        .= a by WAYBEL_1:3
        .= f.(p,q) by A87,Def10;
      hence thesis by A87;
    end;
    suppose
A88:  p = {{{A}}} & q = {A} & u = {A};
      then f.(p,q) "\/" f.(q,u) = (a"\/"b)"\/" f.(q,u) by Def10
        .= Bottom L"\/"(a "\/" b) by A88,Def10
        .= a"\/"b by WAYBEL_1:3
        .= f.(p,q) by A88,Def10;
      hence thesis by A88;
    end;
    suppose
A89:  p = {{{A}}} & q = {A} & u = {{A}};
      then
A90:  f.(p,u) = a by Def10;
      f.(p,q) "\/" f.(q,u) = (a"\/"b)"\/" f.(q,u) by A89,Def10
        .= (a"\/"b) "\/" b by A89,Def10
        .= a"\/"(b"\/"b) by LATTICE3:14
        .= a"\/"b by YELLOW_5:1;
      hence thesis by A90,YELLOW_0:22;
    end;
    suppose
A91:  p = {{{A}}} & q = {A} & u = {{{A}}};
      Bottom L <= f.(p,q) "\/" f.(q,u) by YELLOW_0:44;
      hence thesis by A91,Def10;
    end;
    suppose
A92:  p = {{{A}}} & q = {{A}} & u = {A};
      then f.(p,q) "\/" f.(q,u) = a"\/" f.(q,u) by Def10
        .= a"\/"b by A92,Def10;
      hence thesis by A92,Def10;
    end;
    suppose
A93:  p = {{{A}}} & q = {{A}} & u = {{A}};
      then f.(p,q) "\/" f.(q,u) = a"\/" f.(q,u) by Def10
        .= Bottom L"\/"a by A93,Def10
        .= a by WAYBEL_1:3
        .= f.(p,q) by A93,Def10;
      hence thesis by A93;
    end;
    suppose
A94:  p = {{{A}}} & q = {{A}} & u = {{{A}}};
      Bottom L <= f.(p,q) "\/" f.(q,u) by YELLOW_0:44;
      hence thesis by A94,Def10;
    end;
    suppose
A95:  p = {{{A}}} & q = {{{A}}} & u = {A};
      then f.(p,q) "\/" f.(q,u) = Bottom L"\/" f.(p,u) by Def10
        .= Bottom L"\/"(a"\/"b) by A95,Def10
        .= a"\/"b by WAYBEL_1:3;
      hence thesis by A95,Def10;
    end;
    suppose
A96:  p = {{{A}}} & q = {{{A}}} & u = {{A}};
      then f.(p,q) "\/" f.(q,u) = Bottom L"\/" f.(p,u) by Def10
        .= Bottom L"\/"a by A96,Def10
        .= a by WAYBEL_1:3;
      hence thesis by A96,Def10;
    end;
    suppose
A97:  p = {{{A}}} & q = {{{A}}} & u = {{{A}}};
      Bottom L <= f.(p,q) "\/" f.(q,u) by YELLOW_0:44;
      hence thesis by A97,Def10;
    end;
  end;
A98: for p,q,u being Element of new_set 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_set A;
    assume that
A99: p in B and
A100: q in A and
A101: u in B;
    reconsider q9 = q as Element of A by A100;
    per cases by A99,A100,A101,ENUMSET1:def 1;
    suppose
      q in A & p = {A} & u = {A};
      then f.(p,u) = Bottom L by Def10;
      hence thesis by YELLOW_0:44;
    end;
    suppose
A102: q in A & p = {A} & u = {{A}};
      then
A103: f.(p,q) "\/" f.(q,u) = f.(p,q)"\/"(d.(q9,x)"\/"a"\/"b) by Def10
        .= f.(p,q)"\/"(d.(q9,x)"\/"(a"\/"b)) by LATTICE3:14
        .= (f.(p,q)"\/"d.(q9,x))"\/"(a"\/"b) by LATTICE3:14
        .= ((f.(p,q)"\/"d.(q9,x))"\/"a)"\/"b by LATTICE3:14;
      f.(p,u) = b by A102,Def10;
      hence thesis by A103,YELLOW_0:22;
    end;
    suppose
A104: q in A & p = {A} & u = {{{A}}};
      then
A105: f.(p,u) = a"\/"b by Def10;
      f.(p,q) "\/" f.(q,u) = (d.(q9,x)"\/"a)"\/"f.(q,u) by A104,Def10
        .= (d.(q9,x)"\/"a)"\/"(d.(q9,y)"\/"b) by A104,Def10
        .= d.(q9,x)"\/"(a"\/"(d.(q9,y)"\/"b)) by LATTICE3:14
        .= d.(q9,x)"\/"(d.(q9,y)"\/"(a"\/"b)) by LATTICE3:14
        .= (d.(q9,x)"\/"d.(q9,y))"\/"(a"\/"b) by LATTICE3:14;
      hence thesis by A105,YELLOW_0:22;
    end;
    suppose
A106: q in A & p = {{A}} & u = {A};
      then
A107: f.(p,q) "\/" f.(q,u) = (d.(q9,x)"\/"a"\/"b)"\/"f.(q,u) by Def10
        .= f.(q,u)"\/"(d.(q9,x)"\/"(a"\/"b)) by LATTICE3:14
        .= (f.(q,u)"\/"d.(q9,x))"\/"(a"\/"b) by LATTICE3:14
        .= ((f.(q,u)"\/"d.(q9,x))"\/"a)"\/"b by LATTICE3:14;
      f.(p,u) = b by A106,Def10;
      hence thesis by A107,YELLOW_0:22;
    end;
    suppose
      q in A & p = {{A}} & u = {{A}};
      then f.(p,u) = Bottom L by Def10;
      hence thesis by YELLOW_0:44;
    end;
    suppose
A108: q in A & p = {{A}} & u = {{{A}}};
      then
A109: f.(p,q) "\/" f.(q,u) = (d.(q9,x)"\/"a"\/"b)"\/"f.(q,u) by Def10
        .= (a"\/"(d.(q9,x)"\/"b))"\/"f.(q,u) by LATTICE3:14
        .= a"\/"((d.(q9,x)"\/"b)"\/"f.(q,u)) by LATTICE3:14;
      f.(p,u) = a by A108,Def10;
      hence thesis by A109,YELLOW_0:22;
    end;
    suppose
A110: q in A & p = {{{A}}} & u = {A};
      then
A111: f.(p,u) = a"\/"b by Def10;
      f.(p,q) "\/" f.(q,u) = (d.(q9,y)"\/"b)"\/"f.(q,u) by A110,Def10
        .= (d.(q9,y)"\/"b)"\/"(d.(q9,x)"\/"a) by A110,Def10
        .= d.(q9,y)"\/"(b"\/"(d.(q9,x)"\/"a)) by LATTICE3:14
        .= d.(q9,y)"\/"(d.(q9,x)"\/"(b"\/"a)) by LATTICE3:14
        .= (d.(q9,y)"\/"d.(q9,x))"\/"(a"\/"b) by LATTICE3:14;
      hence thesis by A111,YELLOW_0:22;
    end;
    suppose
A112: q in A & p = {{{A}}} & u = {{A}};
      then
A113: f.(p,q) "\/" f.(q,u) = f.(p,q)"\/"(d.(q9,x)"\/"a"\/"b) by Def10
        .= f.(p,q)"\/"(d.(q9,x)"\/"(a"\/"b)) by LATTICE3:14
        .= (f.(p,q)"\/"d.(q9,x))"\/"(a"\/"b) by LATTICE3:14
        .= ((f.(p,q)"\/"d.(q9,x))"\/"b)"\/"a by LATTICE3:14;
      f.(p,u) = a by A112,Def10;
      hence thesis by A113,YELLOW_0:22;
    end;
    suppose
      q in A & p = {{{A}}} & u = {{{A}}};
      then f.(p,u) = Bottom L by Def10;
      hence thesis by YELLOW_0:44;
    end;
  end;
A114: for p,q,u being Element of new_set 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_set A;
    assume that
A115: p in A and
A116: q in B & u in B;
    reconsider p9 = p as Element of A by A115;
    per cases by A115,A116,ENUMSET1:def 1;
    suppose
A117: p in A & q = {A} & u = {A};
      then f.(p,q) "\/" f.(q,u) = Bottom L"\/"f.(p,q) by Def10
        .= f.(p,q) by WAYBEL_1:3;
      hence thesis by A117;
    end;
    suppose
A118: p in A & q = {A} & u = {{A}};
      then f.(p,u) = d.(p9,x)"\/"a"\/"b by Def10
        .= f.(p,q)"\/"b by A118,Def10;
      hence thesis by A118,Def10;
    end;
    suppose
A119: p in A & q = {A} & u = {{{A}}};
      d.(p9,y) <= d.(p9,x)"\/"d.(x,y) by A2;
      then d.(p9,y)"\/"b <= (d.(p9,x)"\/"d.(x,y))"\/"b by WAYBEL_1:2;
      then
A120: f.(p,u) <= (d.(p9,x)"\/"d.(x,y))"\/"b by A119,Def10;
      d.(p9,x)"\/"b <= d.(p9,x)"\/"b;
      then
      (d.(p9,x)"\/"d.(x,y))"\/"b = (d.(p9,x)"\/"b)"\/"d.(x,y) & (d.(p9,x)
      "\/"b) "\/"d.(x,y) <= (d.(p9,x)"\/"b)"\/"(a"\/" b) by A14,LATTICE3:14
,YELLOW_3:3;
      then
A121: f.(p,u) <= (d.(p9,x)"\/"b)"\/"(a"\/"b) by A120,ORDERS_2:3;
A122: (d.(p9,x)"\/"b)"\/"(a"\/"b) = d.(p9,x)"\/"((b"\/"a)"\/"b) by LATTICE3:14
        .= d.(p9,x)"\/"(a"\/"(b"\/"b)) by LATTICE3:14
        .= d.(p9,x)"\/"(a"\/"b) by YELLOW_5:1
        .= d.(p9,x)"\/"((a"\/"a)"\/"b) by YELLOW_5:1
        .= d.(p9,x)"\/"(a"\/"(a"\/"b)) by LATTICE3:14
        .= (d.(p9,x)"\/"a)"\/"(a"\/"b) by LATTICE3:14;
      f.(p,q) = d.(p9,x)"\/"a by A119,Def10;
      hence thesis by A119,A121,A122,Def10;
    end;
    suppose
A123: p in A & q = {{A}} & u = {A};
      then f.(p,q) = d.(p9,x)"\/"a"\/"b by Def10
        .= f.(p,u)"\/"b by A123,Def10;
      then f.(p,q) "\/" f.(q,u) = (f.(p,u)"\/"b)"\/"b by A123,Def10
        .= f.(p,u)"\/"(b"\/"b) by LATTICE3:14
        .= f.(p,u)"\/"b by YELLOW_5:1;
      hence thesis by YELLOW_0:22;
    end;
    suppose
A124: p in A & q = {{A}} & u = {{A}};
      then f.(p,q) "\/" f.(q,u) = Bottom L"\/"f.(p,q) by Def10
        .= f.(p,q) by WAYBEL_1:3;
      hence thesis by A124;
    end;
    suppose
A125: p in A & q = {{A}} & u = {{{A}}};
      d.(p9,y) <= d.(p9,x)"\/"d.(x,y) by A2;
      then d.(p9,y)"\/"b <= (d.(p9,x)"\/"d.(x,y))"\/"b by WAYBEL_1:2;
      then
A126: f.(p,u) <= (d.(p9,x)"\/"d.(x,y))"\/"b by A125,Def10;
      d.(p9,x)"\/"b <= d.(p9,x)"\/"b;
      then
      (d.(p9,x)"\/"d.(x,y))"\/"b = (d.(p9,x)"\/"b)"\/"d.(x,y) & (d.(p9,x)
      "\/"b) "\/"d.(x,y) <= (d.(p9,x)"\/"b)"\/"(a"\/" b) by A14,LATTICE3:14
,YELLOW_3:3;
      then
A127: f.(p,u) <= (d.(p9,x)"\/"b)"\/"(a"\/"b) by A126,ORDERS_2:3;
A128: (d.(p9,x)"\/"b)"\/"(a"\/"b) = d.(p9,x)"\/"((b"\/"a)"\/"b) by LATTICE3:14
        .= d.(p9,x)"\/"(a"\/"(b"\/"b)) by LATTICE3:14
        .= d.(p9,x)"\/"(a"\/"b) by YELLOW_5:1
        .= d.(p9,x)"\/"((a"\/"a)"\/"b) by YELLOW_5:1
        .= d.(p9,x)"\/"(a"\/"(a"\/"b)) by LATTICE3:14
        .= (d.(p9,x)"\/"(a"\/"b))"\/"a by LATTICE3:14
        .= (d.(p9,x)"\/"a"\/"b)"\/"a by LATTICE3:14;
      f.(p,q) = d.(p9,x)"\/"a"\/"b by A125,Def10;
      hence thesis by A125,A127,A128,Def10;
    end;
    suppose
A129: p in A & q = {{{A}}} & u = {A};
      d.(p9,x) <= d.(p9,y)"\/"d.(y,x) by A2;
      then d.(p9,x)"\/"a <= (d.(p9,y)"\/"d.(y,x))"\/"a by WAYBEL_1:2;
      then
A130: f.(p,u) <= (d.(p9,y)"\/"d.(y,x))"\/"a by A129,Def10;
      d.(y,x) <= a"\/"b & d.(p9,y)"\/"a <= d.(p9,y)"\/"a by A1,A14;
      then
      (d.(p9,y)"\/"d.(y,x))"\/"a = (d.(p9,y)"\/"a)"\/"d.(y,x) & (d.(p9,y)
"\/"a) "\/"d.(y,x) <= (d.(p9,y)"\/"a)"\/"(a"\/" b) by LATTICE3:14,YELLOW_3:3;
      then
A131: f.(p,u) <= (d.(p9,y)"\/"a)"\/"(a"\/"b) by A130,ORDERS_2:3;
A132: (d.(p9,y)"\/"a)"\/"(a"\/"b) = (d.(p9,y)"\/"a"\/"a)"\/"b by LATTICE3:14
        .=(d.(p9,y)"\/"(a"\/"a))"\/" b by LATTICE3:14
        .= (d.(p9,y)"\/"a)"\/"b by YELLOW_5:1
        .= d.(p9,y)"\/"(a"\/"b) by LATTICE3:14
        .= d.(p9,y)"\/"(a"\/"(b"\/"b)) by YELLOW_5:1
        .= d.(p9,y)"\/"((a"\/"b)"\/"b) by LATTICE3:14
        .= (d.(p9,y)"\/"b)"\/"(a"\/"b) by LATTICE3:14;
      f.(p,q) = d.(p9,y)"\/"b by A129,Def10;
      hence thesis by A129,A131,A132,Def10;
    end;
    suppose
A133: p in A & q = {{{A}}} & u = {{A}};
      d.(p9,x) <= d.(p9,y)"\/"d.(y,x) by A2;
      then
A134: d.(p9,x)"\/"(a"\/"b) <= (d.(p9,y)"\/"d.(y,x))"\/"(a"\/"b) by WAYBEL_1:2;
      f.(p,u) = d.(p9,x)"\/"a"\/"b by A133,Def10;
      then
A135: f.(p,u) <= (d.(p9,y)"\/"d.(y,x))"\/"(a"\/"b) by A134,LATTICE3:14;
A136: (d.(p9,y)"\/"a)"\/"(a"\/"b) = (d.(p9,y)"\/"a"\/"a)"\/"b by LATTICE3:14
        .=(d.(p9,y)"\/"(a"\/"a))"\/" b by LATTICE3:14
        .= (d.(p9,y)"\/"a)"\/"b by YELLOW_5:1
        .= (d.(p9,y)"\/"b)"\/" a by LATTICE3:14;
A137: f.(p,q) = d.(p9,y)"\/"b by A133,Def10;
A138: d.(p9,y)"\/"(a"\/"b) <= d.(p9,y)"\/"(a"\/"b);
      d.(y,x) <= a"\/"b & (d.(p9,y)"\/"d.(y,x))"\/"(a"\/"b) = (d.(p9,y)
      "\/"(a"\/"b ))"\/" d.(y,x) by A1,A14,LATTICE3:14;
      then
A139: (d.(p9,y)"\/"d.(y,x))"\/"(a"\/"b)<=(d.(p9,y)"\/"(a"\/"b))"\/"( a
      "\/" b) by A138,YELLOW_3:3;
      (d.(p9,y)"\/"a)"\/"(a"\/"b) = (d.(p9,y)"\/"a)"\/"((a"\/"b)"\/"(a
      "\/" b)) by YELLOW_5:1
        .= d.(p9,y)"\/"(a"\/"((a"\/"b)"\/"(a"\/"b))) by LATTICE3:14
        .= d.(p9,y)"\/"((a"\/"(a"\/"b))"\/"(a"\/"b)) by LATTICE3:14
        .= d.(p9,y)"\/"(((a"\/"a)"\/"b)"\/"(a"\/"b)) by LATTICE3:14
        .= d.(p9,y)"\/"((a"\/"b)"\/"(a"\/"b)) by YELLOW_5:1
        .= (d.(p9,y)"\/"(a"\/"b))"\/"(a"\/"b) by LATTICE3:14;
      then f.(p,u) <= (d.(p9,y)"\/"a)"\/"(a"\/"b) by A139,A135,ORDERS_2:3;
      hence thesis by A133,A137,A136,Def10;
    end;
    suppose
A140: p in A & q = {{{A}}} & u = {{{A}}};
      then f.(p,q) "\/" f.(q,u) = Bottom L"\/"f.(p,q) by Def10
        .= f.(p,q) by WAYBEL_1:3;
      hence thesis by A140;
    end;
  end;
A141: for p,q,u being Element of new_set 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_set A;
    assume p in A & q in A & u in A;
    then reconsider p9 = p, q9 = q, u9 = u as Element of A;
A142: f.(q,u) = d.(q9,u9) by Def10;
    f.(p,u) = d.(p9,u9) & f.(p,q) = d.(p9,q9) by Def10;
    hence thesis by A2,A142;
  end;
  for p,q,u being Element of new_set A holds f.(p,u) <= f.(p,q) "\/" f.( q,u)
  proof
    let p,q,u be Element of new_set A;
    per cases by XBOOLE_0:def 3;
    suppose
      p in A & q in A & u in A;
      hence thesis by A141;
    end;
    suppose
      p in A & q in A & u in B;
      hence thesis by A54;
    end;
    suppose
      p in A & q in B & u in A;
      hence thesis by A3;
    end;
    suppose
      p in A & q in B & u in B;
      hence thesis by A114;
    end;
    suppose
      p in B & q in A & u in A;
      hence thesis by A42;
    end;
    suppose
      p in B & q in A & u in B;
      hence thesis by A98;
    end;
    suppose
      p in B & q in B & u in A;
      hence thesis by A15;
    end;
    suppose
      p in B & q in B & u in B;
      hence thesis by A65;
    end;
  end;
  hence thesis;
end;
