reserve a, b, r, s for Real;

theorem Th45:
  for T being 1-sorted, F being finite Subset-Family of T for F1
being Subset-Family of T st F is Cover of T & F1 = F \ {X where X is Subset of
T: X in F & ex Y being Subset of T st Y in F & X c< Y} holds F1 is Cover of T
proof
  let T be 1-sorted, F be finite Subset-Family of T, F1 be Subset-Family of T
  such that
A1: the carrier of T c= union F and
A2: F1 = F \ {X where X is Subset of T: X in F & ex Y being Subset of T
  st Y in F & X c< Y};
  set ZAW = {X where X is Subset of T: X in F & ex Y being Subset of T st Y in
  F & X c< Y};
  thus the carrier of T c= union F1
  proof
    let t be object;
    assume t in the carrier of T;
    then consider Z being set such that
A3: t in Z and
A4: Z in F by A1,TARSKI:def 4;
    set ALL = {X where X is Subset of T: Z c< X & X in F};
    per cases;
    suppose
A5:   ALL is empty;
      now
        assume Z in ZAW;
        then consider X being Subset of T such that
A6:     Z = X and
        X in F and
A7:     ex Y being Subset of T st Y in F & X c< Y;
        consider Y being Subset of T such that
A8:     Y in F & X c< Y by A7;
        Y in ALL by A6,A8;
        hence contradiction by A5;
      end;
      then Z in F1 by A2,A4,XBOOLE_0:def 5;
      hence thesis by A3,TARSKI:def 4;
    end;
    suppose
      ALL is non empty;
      then consider w being object such that
A9:   w in ALL;
      consider R being Relation of ALL such that
A10:  for x, y being set holds [x,y] in R iff x in ALL & y in ALL & P
      [x,y] from RELSET_1:sch 5;
A11:  R is_reflexive_in ALL
      by A10;
      then
A12:  field R = ALL by ORDERS_1:13;
A13:  R partially_orders ALL
      proof
        thus R is_reflexive_in ALL by A11;
        thus R is_transitive_in ALL
        proof
          let x, y, z be object;
          assume that
A14:      x in ALL and
          y in ALL and
A15:      z in ALL;
          assume
A16:  [x,y] in R & [y,z] in R;
          reconsider x,y,z as set by TARSKI:1;
          x c= y & y c= z by A10,A16;
          then x c= z;
          hence thesis by A10,A14,A15;
        end;
         let x, y be object;
        assume that
        x in ALL and
        y in ALL;
        assume
A17:       [x,y] in R & [y,x] in R;
         reconsider x,y as set by TARSKI:1;
         x c= y & y c= x by A10,A17;
       hence thesis by XBOOLE_0:def 10;
      end;
A18:  R is reflexive by A11,A12;
      ALL has_upper_Zorn_property_wrt R
      proof
        let Y be set such that
A19:    Y c= ALL and
A20:    R |_2 Y is being_linear-order;
        per cases;
        suppose
A21:      Y is non empty;
          defpred U[set] means $1 is non empty & $1 c= Y implies union $1 in Y;
          take union Y;
A22:      U[{}];
A23:      for A, B being set st A in Y & B in Y holds A \/ B in Y
          proof
A24:        R |_2 Y c= R by XBOOLE_1:17;
            R |_2 Y is connected by A20;
            then
A25:        R |_2 Y is_connected_in field (R |_2 Y);
            let A, B be set such that
A26:        A in Y & B in Y;
            field(R |_2 Y) = Y by A12,A18,A19,ORDERS_1:71;
            then [A,B] in R |_2 Y or [B,A] in R |_2 Y or A = B by A26,A25;
            then A c= B or B c= A by A10,A24;
            hence thesis by A26,XBOOLE_1:12;
          end;
A27:      for x, B being set st x in Y & B c= Y & U[B] holds U[B \/ {x}]
          proof
            let x, B be set such that
A28:        x in Y and
A29:        B c= Y & U[B] and
            B \/ {x} is non empty and
            B \/ {x} c= Y;
A30:        union {x} = x by ZFMISC_1:25;
            per cases;
            suppose
              B is empty;
              hence thesis by A28,ZFMISC_1:25;
            end;
            suppose
              B is non empty;
              then x \/ union B in Y by A23,A28,A29;
              hence thesis by A30,ZFMISC_1:78;
            end;
          end;
          consider y being object such that
A31:      y in Y by A21;
          y in ALL by A19,A31;
          then consider X being Subset of T such that
A32:      X = y and
A33:      Z c< X and
          X in F;
A34:      X c= union Y by A31,A32,ZFMISC_1:74;
          then
A35:      Z <> union Y by A33,XBOOLE_0:def 8;
          Z c= X by A33;
          then Z c= union Y by A34;
          then
A36:      Z c< union Y by A35;
A37:      ALL c= F
          proof
            let x be object;
            assume x in ALL;
            then ex X being Subset of T st X = x & Z c< X & X in F;
            hence thesis;
          end;
          then
A38:      Y c= F by A19;
A39:      Y is finite by A19,A37;
          U[Y] from FINSET_1:sch 2(A39,A22,A27);
          then union Y in F by A21,A38;
          hence
A40:      union Y in ALL by A36;
          let z be set;
          assume
A41:      z in Y;
          then P[z,union Y] by ZFMISC_1:74;
          hence thesis by A10,A19,A40,A41;
        end;
        suppose
A42:      Y is empty;
           reconsider w as set by TARSKI:1;
          take w;
          thus w in ALL by A9;
          thus thesis by A42;
        end;
      end;
      then consider M being set such that
A43:  M is_maximal_in R by A12,A13,ORDERS_1:63;
A44:  M in field R by A43;
      then
A45:  ex X being Subset of T st X = M & Z c< X & X in F by A12;
      now
        assume M in ZAW;
        then consider H being Subset of T such that
A46:    M = H and
        H in F and
A47:    ex Y being Subset of T st Y in F & H c< Y;
        consider Y being Subset of T such that
A48:    Y in F and
A49:    H c< Y by A47;
        Z c< Y by A45,A46,A49,XBOOLE_1:56;
        then
A50:    Y in ALL by A48;
        H c= Y by A49;
        then [M,Y] in R by A10,A12,A44,A46,A50;
        hence contradiction by A12,A43,A46,A49,A50;
      end;
      then
A51:  M in F1 by A2,A45,XBOOLE_0:def 5;
      Z c= M by A45;
      hence thesis by A3,A51,TARSKI:def 4;
    end;
  end;
end;
