reserve D for non empty set;
reserve f1,f2,f3,f4,f5,f6,f7,f8,f9,f10 for BinominativeFunction of D;
reserve p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11 for PartialPredicate of D;
reserve q1,q2,q3,q4,q5,q6,q7,q8,q9,q10 for total PartialPredicate of D;
reserve n,m,N for Nat;
reserve fD for PFuncs(D,D)-valued FinSequence;
reserve fB for PFuncs(D,BOOLEAN)-valued FinSequence;
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,v for object;
reserve size for non zero Nat;
reserve inp,pos for FinSequence;
reserve prg for non empty FPrg(ND(V,A))-valued FinSequence;

theorem Th1:
  <*p1,f1,p2*> is SFHT of D & <*p2,f2,p3*> is SFHT of D &
  <*p3,f3,p4*> is SFHT of D & <*p4,f4,p5*> is SFHT of D &
  <*p5,f5,p6*> is SFHT of D & <*p6,f6,p7*> is SFHT of D &
  <*p7,f7,p8*> is SFHT of D &
  <*PP_inversion(p2),f2,p3*> is SFHT of D &
  <*PP_inversion(p3),f3,p4*> is SFHT of D &
  <*PP_inversion(p4),f4,p5*> is SFHT of D &
  <*PP_inversion(p5),f5,p6*> is SFHT of D &
  <*PP_inversion(p6),f6,p7*> is SFHT of D &
  <*PP_inversion(p7),f7,p8*> is SFHT of D
  implies <*p1,PP_composition(f1,f2,f3,f4,f5,f6,f7),p8*> is SFHT of D
  proof
    assume that
A1: <*p1,f1,p2*> is SFHT of D and
A2: <*p2,f2,p3*> is SFHT of D and
A3: <*p3,f3,p4*> is SFHT of D and
A4: <*p4,f4,p5*> is SFHT of D and
A5: <*p5,f5,p6*> is SFHT of D and
A6: <*p6,f6,p7*> is SFHT of D and
A7: <*p7,f7,p8*> is SFHT of D and
A8: <*PP_inversion(p2),f2,p3*> is SFHT of D and
A9: <*PP_inversion(p3),f3,p4*> is SFHT of D and
A10: <*PP_inversion(p4),f4,p5*> is SFHT of D and
A11: <*PP_inversion(p5),f5,p6*> is SFHT of D and
A12: <*PP_inversion(p6),f6,p7*> is SFHT of D and
A13: <*PP_inversion(p7),f7,p8*> is SFHT of D;
    <*p1,PP_composition(f1,f2,f3,f4,f5,f6),p7*>
    is SFHT of D by A1,A2,A3,A4,A5,A6,A8,A9,A10,A11,A12,NOMIN_7:3;
    hence thesis by A7,A13,NOMIN_3:25;
  end;
