reserve D for non empty set;
reserve f1,f2,f3,f4,f5 for BinominativeFunction of D;
reserve p,q,r,t,w,u for PartialPredicate of D;

theorem Th1:
  <*p,f1,q*> is SFHT of D & <*q,f2,r*> is SFHT of D &
  <*r,f3,w*> is SFHT of D & <*w,f4,t*> is SFHT of D &
  <*t,f5,u*> is SFHT of D &
  <*PP_inversion(q),f2,r*> is SFHT of D &
  <*PP_inversion(r),f3,w*> is SFHT of D &
  <*PP_inversion(w),f4,t*> is SFHT of D &
  <*PP_inversion(t),f5,u*> is SFHT of D
  implies <*p,PP_composition(f1,f2,f3,f4,f5),u*> is SFHT of D
  proof
    assume that
A1: <*p,f1,q*> is SFHT of D and
A2: <*q,f2,r*> is SFHT of D and
A3: <*r,f3,w*> is SFHT of D and
A4: <*w,f4,t*> is SFHT of D and
A5: <*t,f5,u*> is SFHT of D and
A6: <*PP_inversion(q),f2,r*> is SFHT of D and
A7: <*PP_inversion(r),f3,w*> is SFHT of D and
A8: <*PP_inversion(w),f4,t*> is SFHT of D and
A9: <*PP_inversion(t),f5,u*> is SFHT of D;
    <*p,PP_composition(f1,f2),r*> is SFHT of D by A1,A2,A6,NOMIN_3:25;
    then <*p,PP_composition(PP_composition(f1,f2),f3),w*> is SFHT of D
    by A3,A7,NOMIN_3:25;
    then <*p,PP_composition(PP_composition(f1,f2,f3),f4),t*>
    is SFHT of D by A4,A8,NOMIN_3:25;
    hence thesis by A5,A9,NOMIN_3:25;
  end;
