reserve k,m,n for Nat,
  a, b, c for object,
  x, y, X, Y, Z for set,
  D for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R, P1, P2, Q1, Q2, R1, R2 for FinSequence-membered set;
reserve S, T for non empty FinSequence-membered set;
reserve A for Function of P, NAT;
reserve U, V, W for Subset of P*;
reserve k,l,m,n,i,j for Nat,
  a, b, c for object,
  x, y, z, X, Y, Z for set,
  D, D1, D2 for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R for FinSequence-membered set;
reserve B, C for antichain;
reserve S, T for Polish-language;
reserve A for Polish-arity-function of T;
reserve U, V, W for Polish-language of T;
reserve F, G for Polish-WFF of T, A;
reserve f for Polish-recursion-function of A, D;
reserve K, K1, K2 for Function of Polish-WFF-set(T, A), D;
reserve L for non trivial Polish-language;
reserve E for Polish-arity-function of L;
reserve g for Polish-recursion-function of E, D;
reserve J, J1, J2, J3 for Subset of Polish-WFF-set(L, E);
reserve H for Function of J, D;
reserve H1 for Function of J1, D;
reserve H2 for Function of J2, D;
reserve H3 for Function of J3, D;

theorem Th79:
  for L, E for t being Element of L for F being Polish-WFF of L, E holds
    Polish-WFF-head F = t
      iff ex u being Element of Polish-WFF-set(L, E)^^(E.t)
        st F = Polish-operation(L, E, t).u
proof
 let L, E;
 let t be Element of L;
 let F be Polish-WFF of L, E;
 set W = Polish-WFF-set(L, E);
 set H = Polish-operation(L, E, t);
 A2: dom H = W^^(E.t) by FUNCT_2:def 1;
 thus Polish-WFF-head F = t implies ex u being Element of W^^(E.t) st F = H.u
 proof
   assume A3: Polish-WFF-head F = t;
   set u = (L, E)-tail F;
   reconsider u as Element of W^^(E.t) by A3;
   take u;
   thus F = t^u by A3 .= H.u by A2, Def12;
 end;
 given u being Element of W^^(E.t) such that A20: F = H.u;
 reconsider u as FinSequence;
 thus Polish-WFF-head F = t by A20;
end;
