let V, A be set ; :: thesis: for z being Element of V
for loc being V -valued Function
for n0 being Nat st not V is empty & A is_without_nonatomicND_wrt V & ( for T being TypeSCNominativeData of V,A holds
( loc /. 1 is_a_value_on T & loc /. 3 is_a_value_on T ) ) holds
<*(PP_and ((Equality (A,(loc /. 1),(loc /. 3))),(factorial_inv (A,loc,n0)))),(SC_assignment ((denaming (V,A,(loc /. 4))),z)),(valid_factorial_output (A,z,n0))*> is SFHT of (ND (V,A))

let z be Element of V; :: thesis: for loc being V -valued Function
for n0 being Nat st not V is empty & A is_without_nonatomicND_wrt V & ( for T being TypeSCNominativeData of V,A holds
( loc /. 1 is_a_value_on T & loc /. 3 is_a_value_on T ) ) holds
<*(PP_and ((Equality (A,(loc /. 1),(loc /. 3))),(factorial_inv (A,loc,n0)))),(SC_assignment ((denaming (V,A,(loc /. 4))),z)),(valid_factorial_output (A,z,n0))*> is SFHT of (ND (V,A))

let loc be V -valued Function; :: thesis: for n0 being Nat st not V is empty & A is_without_nonatomicND_wrt V & ( for T being TypeSCNominativeData of V,A holds
( loc /. 1 is_a_value_on T & loc /. 3 is_a_value_on T ) ) holds
<*(PP_and ((Equality (A,(loc /. 1),(loc /. 3))),(factorial_inv (A,loc,n0)))),(SC_assignment ((denaming (V,A,(loc /. 4))),z)),(valid_factorial_output (A,z,n0))*> is SFHT of (ND (V,A))

let n0 be Nat; :: thesis: ( not V is empty & A is_without_nonatomicND_wrt V & ( for T being TypeSCNominativeData of V,A holds
( loc /. 1 is_a_value_on T & loc /. 3 is_a_value_on T ) ) implies <*(PP_and ((Equality (A,(loc /. 1),(loc /. 3))),(factorial_inv (A,loc,n0)))),(SC_assignment ((denaming (V,A,(loc /. 4))),z)),(valid_factorial_output (A,z,n0))*> is SFHT of (ND (V,A)) )

set s = loc /. 4;
<*(SC_Psuperpos ((valid_factorial_output (A,z,n0)),(denaming (V,A,(loc /. 4))),z)),(SC_assignment ((denaming (V,A,(loc /. 4))),z)),(valid_factorial_output (A,z,n0))*> is SFHT of (ND (V,A)) by NOMIN_3:29;
hence ( not V is empty & A is_without_nonatomicND_wrt V & ( for T being TypeSCNominativeData of V,A holds
( loc /. 1 is_a_value_on T & loc /. 3 is_a_value_on T ) ) implies <*(PP_and ((Equality (A,(loc /. 1),(loc /. 3))),(factorial_inv (A,loc,n0)))),(SC_assignment ((denaming (V,A,(loc /. 4))),z)),(valid_factorial_output (A,z,n0))*> is SFHT of (ND (V,A)) ) by ; :: thesis: verum