reserve i for Nat, x,y for set;
reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;

theorem Th52:
  for M,O,N,I being set st I in M & N in M
  ex C being 4-connectives non empty non void strict ConnectivesSignature st
  C is (1,I,N)-array 1-1-connectives &
  M c= the carrier of C & O misses the carrier' of C &
  (the ResultSort of C).((the connectives of C).2) nin M
  proof
    let M,O,N,I be set;
    assume A1: I in M & N in M;
    set X = succ M; set Y = {O, succ O, succ succ O, succ succ succ O};
    reconsider o0 = O, o1 = succ O, o2 = succ succ O,
    o3 = succ succ succ O as Element of Y by ENUMSET1:def 2;
    reconsider m = M, i = I, n = N as Element of X
    by A1,XBOOLE_0:def 3,ORDINAL1:6;
    set A = (o0,o1,o2,o3)-->(<*m,n*>,<*m,n,i*>,<*m*>,<*n,i*>);
A3: o0 in o1 & o1 in o2 & o2 in o3 by ORDINAL1:6; then
A4: o0,o1,o2,o3 are_mutually_distinct by XREGULAR:7; then
    rng A = {<*m,n*>,<*m,n,i*>,<*m*>,<*n,i*>} & <*m,n*> in X* &
    <*m,n,i*> in X* &
    <*m*> in X* & <*n,i*> in X* by FUNCT_4:143,FINSEQ_1:def 11; then
    dom A = Y & rng A c= X* by FUNCT_4:137,QUATERNI:5; then
    reconsider A as Function of Y,X* by FUNCT_2:2;
    set R = (o0,o1,o2,o3)-->(i,m,n,m);
    rng R = {i,m,n,m} by A4,FUNCT_4:143; then
    dom R = Y & rng R c= X by FUNCT_4:137,QUATERNI:5; then
    reconsider R as Function of Y,X by FUNCT_2:2;
    set c = <*o0,o1,o2,o3*>;
    set C = ConnectivesSignature(#X,Y,A,R,c#);
    C is 4-connectives
    by CARD_1:def 7; then
    reconsider C as 4-connectives non empty non void strict
    ConnectivesSignature;
    take C;
    thus C is (1,I,N)-array
    proof
      thus len the connectives of C >= 1+3 by CARD_1:def 7;
      reconsider K = n, L = i, J = m as Element of C;
      take J,K,L;
      thus L = I & K = N;
      thus J <> L & J <> K by A1;
      c.1 = o0 & o0 <> o1 & o0 <> o2 & o0 <> o3 by A3,XREGULAR:7;
      hence (the Arity of C).((the connectives of C).1) = <*J,K*> &
      (the ResultSort of C).((the connectives of C).1) = L
         by FUNCT_4:142;
      c.2 = o1 & o2 <> o1 & o1 <> o3 by A3;
      hence (the Arity of C).((the connectives of C).(1+1)) = <*J,K,L*> &
      (the ResultSort of C).((the connectives of C).(1+1)) = J
        by FUNCT_4:141;
      c.3 = o2 & o2 <> o3 by A3;
      hence (the Arity of C).((the connectives of C).(1+2)) = <*J*> &
      (the ResultSort of C).((the connectives of C).(1+2)) = K
        by FUNCT_4:140;
      thus (the Arity of C).((the connectives of C).(1+3)) = <*K,L*> &
      (the ResultSort of C).((the connectives of C).(1+3)) = J
        by FUNCT_4:139;
    end;
    thus the connectives of C is one-to-one by A4,Th14;
    thus M c= the carrier of C by XBOOLE_1:7;
    now
      given x being object such that
A5:   x in O & x in the carrier' of C;
      x = o0 or x = o1 or x = o2 or x = o3 by A5,ENUMSET1:def 2;
      hence contradiction by A5,A3,XREGULAR:7,8;
    end;
    hence O misses the carrier' of C by XBOOLE_0:3;
    c.2 = o1 & o2 <> o1 & o1 <> o3 by A3;
    then A: (the ResultSort of C).((the connectives of C).(1+1)) = m
            by FUNCT_4:141;
    reconsider nn = (the ResultSort of C).((the connectives of C).2)
      as set;
    not nn in nn;
    hence thesis by A;
  end;
