reserve D for non empty set;
reserve m,n,N for Nat;
reserve size for non zero Nat;
reserve f1,f2,f3,f4,f5,f6 for BinominativeFunction of D;
reserve p1,p2,p3,p4,p5,p6,p7 for PartialPredicate of D;
reserve d,v for object;
reserve V,A for set;
reserve z for Element of V;
reserve val for Function;
reserve loc for V-valued Function;
reserve d1 for NonatomicND of V,A;
reserve T for TypeSCNominativeData of V,A;
reserve n0 for Nat;

theorem Th20:
  V is non empty & A is_without_nonatomicND_wrt V &
  (for T holds loc/.1 is_a_value_on T & loc/.3 is_a_value_on T)
  implies
  PP_and(Equality(A,loc/.1,loc/.3),Fibonacci_inv(A,loc,n0))
  ||=
  SC_Psuperpos(valid_Fibonacci_output(A,z,n0),denaming(V,A,loc/.4),z)
  proof
    set i = loc/.1, j = loc/.2, n = loc/.3, s = loc/.4, b = loc/.5, c = loc/.6;
    set D = ND(V,A);
    set inv = Fibonacci_inv(A,loc,n0);
    set Di = denaming(V,A,i);
    set Dn = denaming(V,A,n);
    set Ds = denaming(V,A,s);
    set Dz = denaming(V,A,z);
    set ass = SC_assignment(Ds,z);
    set out = valid_Fibonacci_output(A,z,n0);
    set p = SC_Psuperpos(out,Ds,z);
    set E = Equality(A,i,n);
    set EM = {i,j,n,s,b,c};
    assume that
A1: V is non empty & A is_without_nonatomicND_wrt V and
A2: for T holds i is_a_value_on T & n is_a_value_on T;
    let d be Element of D such that
A3: d in dom PP_and(E,inv) and
A4: (PP_and(E,inv)).d = TRUE;
A5: dom p = {d where d is TypeSCNominativeData of V,A:
      local_overlapping(V,A,d,Ds.d,z) in dom out & d in dom Ds}
      by NOMIN_2:def 11;
A6: dom out = {d where d is TypeSCNominativeData of V,A: d in dom Dz}
    by Def17;
A7: dom Ds = {d where d is NonatomicND of V,A: s in dom d} by NOMIN_1:def 18;
A8: dom Dz = {d where d is NonatomicND of V,A: z in dom d} by NOMIN_1:def 18;
A9: d in dom E by A3,A4,PARTPR_1:23;
A10: d in dom inv by A3,A4,PARTPR_1:23;
A11: dom E = dom Di /\ dom Dn by A2,NOMIN_4:11;
     then
A12: d in dom Di by A9,XBOOLE_0:def 4;
     inv.d = TRUE by A3,A4,PARTPR_1:23;
     then Fibonacci_inv_pred A,loc,n0,d by A10,Def19;
     then consider d1 being NonatomicND of V,A such that
A13: d = d1 and
A14: EM c= dom d1 and
     d1.j = 1 and
A15: d1.n = n0 and
A16: ex I being Nat st I = d1.i & d1.s = Fib(I) & d1.b = Fib(I+1);
A17: i in EM by ENUMSET1:def 4;
A18: n in EM by ENUMSET1:def 4;
A19: s in EM by ENUMSET1:def 4;
     reconsider dd = d as TypeSCNominativeData of V,A by NOMIN_1:39;
     set L = local_overlapping(V,A,dd,Ds.dd,z);
A20: dd in dom Ds by A7,A13,A14,A19;
     then Ds.d1 in D by A13,PARTFUN1:4;
     then
A21: ex d2 being TypeSCNominativeData of V,A st Ds.d1 = d2;
     then
A22: L in dom Dz by A1,A13,NOMIN_4:6;
     then
A23: L in dom out by A6;
     hence
A24: d in dom p by A5,A20;
     valid_Fibonacci_output_pred A,z,n0,L
     proof
       consider I being Nat such that
A25:   I = d1.i and
A26:   d1.s = Fib(I) and
       d1.b = Fib(I+1) by A16;
A27:   ex d being NonatomicND of V,A st L = d & z in dom d by A8,A22;
       then reconsider dlo = L as NonatomicND of V,A;
       take dlo;
       thus L = dlo;
       thus z in dom dlo by A27;
A28:   i is_a_value_on dd by A2;
A29:   n is_a_value_on dd by A2;
A30:   dom <:Di,Dn:> = dom Di /\ dom Dn by FUNCT_3:def 7;
       d in dom <:Di,Dn:> by A9,A11,FUNCT_3:def 7;
       then
A31:   E.d = (Equality(A)).(<:Di,Dn:>.d) by FUNCT_1:13;
A32:   d in dom Dn by A9,A11,XBOOLE_0:def 4;
A33:   <:Di,Dn:>.d = [Di.d,Dn.d] by A9,A11,A30,FUNCT_3:def 7;
A34:   Di.d = denaming(i,d1) by A13,A12,NOMIN_1:def 18
       .= d1.i by A14,A17,NOMIN_1:def 12;
A35:   Dn.d = denaming(n,d1) by A13,A32,NOMIN_1:def 18
       .= d1.n by A14,A18,NOMIN_1:def 12;
A36:   Ds.d = denaming(s,d1) by A20,A13,NOMIN_1:def 18
       .= d1.s by A14,A19,NOMIN_1:def 12;
       (Equality(A)).(Di.d,Dn.d) <> FALSE by A3,A4,A31,A33,PARTPR_1:23;
       then Di.d = Dn.d by A28,A29,NOMIN_4:def 9;
       hence dlo.z = Fib(n0) by A1,A13,A15,A21,A25,A26,A34,A35,A36,NOMIN_2:10;
     end;
     hence TRUE = out.L by A23,Def17
     .= p.d by A24,NOMIN_2:35;
   end;
