let V, A be set ; :: thesis: for loc being V -valued Function
for z being Element of V
for x0, y0, p0, q0 being Integer
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))),(Lucas_inv (A,loc,x0,y0,p0,q0,n0)))),(SC_assignment ((denaming (V,A,(loc /. 4))),z)),(valid_Lucas_output (A,z,x0,y0,p0,q0,n0))*> is SFHT of (ND (V,A))
let loc be V -valued Function; :: thesis: for z being Element of V
for x0, y0, p0, q0 being Integer
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))),(Lucas_inv (A,loc,x0,y0,p0,q0,n0)))),(SC_assignment ((denaming (V,A,(loc /. 4))),z)),(valid_Lucas_output (A,z,x0,y0,p0,q0,n0))*> is SFHT of (ND (V,A))
let z be Element of V; :: thesis: for x0, y0, p0, q0 being Integer
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))),(Lucas_inv (A,loc,x0,y0,p0,q0,n0)))),(SC_assignment ((denaming (V,A,(loc /. 4))),z)),(valid_Lucas_output (A,z,x0,y0,p0,q0,n0))*> is SFHT of (ND (V,A))
let x0, y0, p0, q0 be Integer; :: 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))),(Lucas_inv (A,loc,x0,y0,p0,q0,n0)))),(SC_assignment ((denaming (V,A,(loc /. 4))),z)),(valid_Lucas_output (A,z,x0,y0,p0,q0,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))),(Lucas_inv (A,loc,x0,y0,p0,q0,n0)))),(SC_assignment ((denaming (V,A,(loc /. 4))),z)),(valid_Lucas_output (A,z,x0,y0,p0,q0,n0))*> is SFHT of (ND (V,A)) )
set s = loc /. 4;
<*(SC_Psuperpos ((valid_Lucas_output (A,z,x0,y0,p0,q0,n0)),(denaming (V,A,(loc /. 4))),z)),(SC_assignment ((denaming (V,A,(loc /. 4))),z)),(valid_Lucas_output (A,z,x0,y0,p0,q0,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))),(Lucas_inv (A,loc,x0,y0,p0,q0,n0)))),(SC_assignment ((denaming (V,A,(loc /. 4))),z)),(valid_Lucas_output (A,z,x0,y0,p0,q0,n0))*> is SFHT of (ND (V,A)) ) by Th19, NOMIN_3:15; :: thesis: verum