reserve x,y,P,Q for Integer;
reserve a,b,n for Nat;
reserve V,A for set;
reserve val for Function;
reserve loc for V-valued Function;
reserve d1 for NonatomicND of V,A;
reserve p for SCPartialNominativePredicate of V,A;
reserve d for object;
reserve z for Element of V;
reserve T for TypeSCNominativeData of V,A;
reserve size for non zero Nat;
reserve x0, y0, p0, q0 for Integer;
reserve n0 for Nat;

theorem Th14:
  for val being 10-element FinSequence holds
  V is non empty & A is_without_nonatomicND_wrt V &
  Seg 10 c= dom loc & loc|Seg 10 is one-to-one & loc,val are_different_wrt 10
  implies
  <* valid_Lucas_input(V,A,val,x0,y0,p0,q0,n0),
     initial_assignments(A,loc,val,10),
     Lucas_inv(A,loc,x0,y0,p0,q0,n0) *> is SFHT of ND(V,A)
  proof
    let val be 10-element FinSequence;
    set size = 10;
    set G = initial_assignments_Seq(A,loc,val,size);
A1: G.1 = SC_assignment(denaming(V,A,val.1),loc/.1) by NOMIN_7:def 8;
    set i = loc/.1, j = loc/.2, n = loc/.3, s = loc/.4, b = loc/.5, c = loc/.6;
    set p = loc/.7, q = loc/.8, ps = loc/.9, qc = loc/.10;
    set i1 = val.1, j1 = val.2, n1 = val.3, s1 = val.4, b1 = val.5, c1 = val.6;
    set p1 = val.7, q1 = val.8, ps1 = val.9, qc1 = val.10;
    set EN = {i1,j1,n1,s1,b1,c1,p1,q1,ps1,qc1};
    set D = ND(V,A);
    set I = valid_Lucas_input(V,A,val,x0,y0,p0,q0,n0);
    set inv = Lucas_inv(A,loc,x0,y0,p0,q0,n0);
    set DS = denamingSeq(V,A,val);
    set asi = SC_assignment(DS.1,i);
    set asj = SC_assignment(DS.2,j);
    set asn = SC_assignment(DS.3,n);
    set ass = SC_assignment(DS.4,s);
    set asb = SC_assignment(DS.5,b);
    set asc = SC_assignment(DS.6,c);
    set asp = SC_assignment(DS.7,p);
    set asq = SC_assignment(DS.8,q);
    set asps = SC_assignment(DS.9,ps);
    set asqc = SC_assignment(DS.10,qc);
    set SE = SC_Psuperpos_Seq(loc,val,inv);
    assume that
A2: V is non empty and
A3: A is_without_nonatomicND_wrt V and
A4: Seg 10 c= dom loc and
A5: loc|Seg 10 is one-to-one and
A6: loc,val are_different_wrt 10;
A7: len val = 10 by CARD_1:def 7;
A8: len SE = 10 by NOMIN_8:def 9,A7;
A9: len DS = 10 by NOMIN_8:def 8,A7;
A10: 2=1+1 & 3=2+1 & 4=3+1 & 5=4+1 & 6=5+1 & 7=6+1 & 8=7+1 & 9=8+1 & 10=9+1;
A11: SE.1 = SC_Psuperpos(inv,denaming(V,A,val.(len val)),loc/.(len val))
    by NOMIN_8:def 9;
A12: SE.2 = SC_Psuperpos(SE.1,denaming(V,A,val.(len val-1)),loc/.(len val-1))
    by A10,A8,NOMIN_8:def 9;
A13: SE.3 = SC_Psuperpos(SE.2,denaming(V,A,val.(len val-2)),loc/.(len val-2))
    by A10,A8,NOMIN_8:def 9;
A14: SE.4 = SC_Psuperpos(SE.3,denaming(V,A,val.(len val-3)),loc/.(len val-3))
    by A10,A8,NOMIN_8:def 9;
A15: SE.5 = SC_Psuperpos(SE.4,denaming(V,A,val.(len val-4)),loc/.(len val-4))
    by A10,A8,NOMIN_8:def 9;
A16: SE.6 = SC_Psuperpos(SE.5,denaming(V,A,val.(len val-5)),loc/.(len val-5))
    by A10,A8,NOMIN_8:def 9;
A17: SE.7 = SC_Psuperpos(SE.6,denaming(V,A,val.(len val-6)),loc/.(len val-6))
    by A10,A8,NOMIN_8:def 9;
A18: SE.8 = SC_Psuperpos(SE.7,denaming(V,A,val.(len val-7)),loc/.(len val-7))
    by A10,A8,NOMIN_8:def 9;
A19: SE.9 = SC_Psuperpos(SE.8,denaming(V,A,val.(len val-8)),loc/.(len val-8))
    by A10,A8,NOMIN_8:def 9;
A20: SE.10 = SC_Psuperpos(SE.9,denaming(V,A,val.(len val-9)),loc/.(len val-9))
    by A10,A8,NOMIN_8:def 9;
A21: DS.1 = denaming(V,A,val.1) by NOMIN_8:def 8,A9;
A22: DS.2 = denaming(V,A,val.2) by NOMIN_8:def 8,A9;
A23: DS.3 = denaming(V,A,val.3) by NOMIN_8:def 8,A9;
A24: DS.4 = denaming(V,A,val.4) by NOMIN_8:def 8,A9;
A25: DS.5 = denaming(V,A,val.5) by NOMIN_8:def 8,A9;
A26: DS.6 = denaming(V,A,val.6) by NOMIN_8:def 8,A9;
A27: DS.7 = denaming(V,A,val.7) by NOMIN_8:def 8,A9;
A28: DS.8 = denaming(V,A,val.8) by NOMIN_8:def 8,A9;
A29: DS.9 = denaming(V,A,val.9) by NOMIN_8:def 8,A9;
A30: DS.10 = denaming(V,A,val.10) by NOMIN_8:def 8,A9;
A31: <*SE.1,asqc,inv*> is SFHT of D by A7,A11,A30,NOMIN_3:29;
A32: <*SE.2,asps,SE.1*> is SFHT of D by A7,A12,A29,NOMIN_3:29;
A33: <*SE.3,asq,SE.2*> is SFHT of D by A7,A13,A28,NOMIN_3:29;
A34: <*SE.4,asp,SE.3*> is SFHT of D by A7,A14,A27,NOMIN_3:29;
A35: <*SE.5,asc,SE.4*> is SFHT of D by A7,A15,A26,NOMIN_3:29;
A36: <*SE.6,asb,SE.5*> is SFHT of D by A7,A16,A25,NOMIN_3:29;
A37: <*SE.7,ass,SE.6*> is SFHT of D by A7,A17,A24,NOMIN_3:29;
A38: <*SE.8,asn,SE.7*> is SFHT of D by A7,A18,A23,NOMIN_3:29;
A39: <*SE.9,asj,SE.8*> is SFHT of D by A7,A19,A22,NOMIN_3:29;
A40: <*SE.10,asi,SE.9*> is SFHT of D by A7,A20,A21,NOMIN_3:29;
A41: G.2 = PP_composition(asi,asj) by A10,A21,A22,A1,NOMIN_7:def 8;
A42: G.3 = PP_composition(G.2,asn) by A10,A23,NOMIN_7:def 8;
A43: G.4 = PP_composition(G.3,ass) by A10,A24,NOMIN_7:def 8;
A44: G.5 = PP_composition(G.4,asb) by A10,A25,NOMIN_7:def 8;
A45: G.6 = PP_composition(G.5,asc) by A10,A26,NOMIN_7:def 8;
A46: G.7 = PP_composition(G.6,asp) by A10,A27,NOMIN_7:def 8;
A47: G.8 = PP_composition(G.7,asq) by A10,A28,NOMIN_7:def 8;
A48: G.9 = PP_composition(G.8,asps) by A10,A29,NOMIN_7:def 8;
A49: initial_assignments(A,loc,val,10) =
    PP_composition(asi,asj,asn,ass,asb,asc,asp,asq,asps,asqc)
    by A10,A30,A41,A42,A43,A44,A45,A46,A47,A48,NOMIN_7:def 8;
    I ||= SE.10 by A2,A3,A4,A5,A6,A8,Th13;
    then
A50: <*I,asi,SE.9*> is SFHT of D by A40,NOMIN_3:15;
A51: <*PP_inversion(SE.9),asj,SE.8*> is SFHT of D by A7,A19,A22,NOMIN_4:16;
A52: <*PP_inversion(SE.8),asn,SE.7*> is SFHT of D by A7,A18,A23,NOMIN_4:16;
A53: <*PP_inversion(SE.7),ass,SE.6*> is SFHT of D by A7,A17,A24,NOMIN_4:16;
A54: <*PP_inversion(SE.6),asb,SE.5*> is SFHT of D by A7,A16,A25,NOMIN_4:16;
A55: <*PP_inversion(SE.5),asc,SE.4*> is SFHT of D by A7,A15,A26,NOMIN_4:16;
A56: <*PP_inversion(SE.4),asp,SE.3*> is SFHT of D by A7,A14,A27,NOMIN_4:16;
A57: <*PP_inversion(SE.3),asq,SE.2*> is SFHT of D by A7,A13,A28,NOMIN_4:16;
A58: <*PP_inversion(SE.2),asps,SE.1*> is SFHT of D by A7,A12,A29,NOMIN_4:16;
    <*PP_inversion(SE.1),asqc,inv*> is SFHT of D by A7,A11,A30,NOMIN_4:16;
    hence thesis by A49,A32,A31,A33,A34,A35,A36,A37,A38,A39,A50,A51,A52,A53,
    A54,A55,A56,A57,A58,NOMIN_8:4;
  end;
