reserve a for set;
reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;
reserve L for complete LATTICE;
reserve AR for Relation of L;
reserve x, y, z for Element of L;

theorem Th50:
  for R being approximating auxiliary Relation of L holds
  x << z & x <> z implies
  ex y being Element of L st [x,y] in R & [y,z] in R & x <> y
proof
  let R be approximating auxiliary Relation of L;
  assume that
A1: x << z and
A2: x <> z;
  set I = {u where u is Element of L : ex y be Element of L st
  [u,y] in R & [y,z] in R };
A3: [Bottom L,Bottom L] in R by Def6;
  [Bottom L,z] in R by Def6;
  then
A4: Bottom L in I by A3;
  I c= the carrier of L
  proof
    let v be object;
    assume v in I;
    then ex u1 being Element of L st v = u1 & ex y be Element of L st
    [u1,y] in R & [y,z] in R;
    hence thesis;
  end;
  then reconsider I as non empty Subset of L by A4;
A5: I is lower
  proof
    let x1,y1 be Element of L;
    assume that
A6: x1 in I and
A7: y1 <= x1;
    consider v1 be Element of L such that
A8: v1 = x1 and
A9: ex s1 be Element of L st [v1,s1] in R & [s1,z] in R by A6;
    consider s1 be Element of L such that
A10: [v1,s1] in R and
A11: [s1,z] in R by A9;
    s1 <= s1;
    then [y1, s1] in R by A7,A8,A10,Def4;
    hence thesis by A11;
  end;
  I is directed
  proof
    let u1,u2 be Element of L;
    assume that
A12: u1 in I and
A13: u2 in I;
    consider v1 be Element of L such that
A14: v1 = u1 and
A15: ex y1 be Element of L st [v1,y1] in R & [y1,z] in R by A12;
    consider v2 be Element of L such that
A16: v2 = u2 and
A17: ex y2 be Element of L st [v2,y2] in R & [y2,z] in R by A13;
    consider y1 be Element of L such that
A18: [v1,y1] in R and
A19: [y1,z] in R by A15;
    consider y2 be Element of L such that
A20: [v2,y2] in R and
A21: [y2,z] in R by A17;
    take u1 "\/" u2;
A22: [u1 "\/" u2, y1 "\/" y2] in R by A14,A16,A18,A20,Th1;
    [y1 "\/" y2, z] in R by A19,A21,Def5;
    hence thesis by A22,YELLOW_0:22;
  end;
  then reconsider I as Ideal of L by A5;
  sup I = z
  proof
    set z9 = sup I;
    assume
A23: z9 <> z;
A24: I c= R-below z
    proof
      let a be object;
      assume a in I;
      then consider u be Element of L such that
A25:  a = u and
A26:  ex y2 be Element of L st [u,y2] in R & [y2,z] in R;
      consider y2 be Element of L such that
A27:  [u,y2] in R and
A28:  [y2,z] in R by A26;
A29:  u <= y2 by A27,Def3;
      z <= z;
      then [u,z] in R by A28,A29,Def4;
      hence thesis by A25;
    end;
A30: ex_sup_of I,L by YELLOW_0:17;
    ex_sup_of (R-below z),L by YELLOW_0:17;
    then
A31: sup I <= sup (R-below z) by A24,A30,YELLOW_0:34;
    z = sup (R-below z) by Def17;
    then z9 < z by A23,A31,ORDERS_2:def 6;
    then not z <= z9 by ORDERS_2:6;
    then consider y be Element of L such that
A32: [y, z] in R and
A33: not y <= z9 by Th48;
    consider u be Element of L such that
A34: [u, y] in R and
A35: not u <= z9 by A33,Th48;
A36: u in I by A32,A34;
    z9 = "\/"(I,L) & ex_sup_of I,L iff z9 is_>=_than I &
    for b being Element of L st b is_>=_than I holds z9 <= b by YELLOW_0:30;
    hence contradiction by A35,A36,YELLOW_0:17;
  end;
  then x in I by A1,WAYBEL_3:20;
  then consider v be Element of L such that
A37: v = x and
A38: ex y9 be Element of L st [v,y9] in R & [y9,z] in R;
  consider y9 be Element of L such that
A39: [v,y9] in R and
A40: [y9,z] in R by A38;
A41: x <= y9 by A37,A39,Def3;
  z <= z;
  then [x,z] in R by A40,A41,Def4;
  then consider y99 be Element of L such that
A42: x <= y99 and
A43: [y99,z] in R and
A44: x <> y99 by A2,Th49;
A45: x < y99 by A42,A44,ORDERS_2:def 6;
  set Y = y9 "\/" y99;
A46: y9 <= Y by YELLOW_0:22;
  y99 <= Y by YELLOW_0:22;
  then
A47: x <> Y by A45,ORDERS_2:7;
  x <= x;
  then
A48: [x,Y] in R by A37,A39,A46,Def4;
  [Y,z] in R by A40,A43,Def5;
  hence thesis by A47,A48;
end;
