let A be set ; :: thesis: for x0, y0, p0, q0 being Integer
for n0 being Nat
for V being non empty set
for loc being b₆ -valued 10 -element FinSequence st A is complex-containing & A is_without_nonatomicND_wrt V & ( for T being TypeSCNominativeData of V,A holds
( loc /. 1 is_a_value_on T & loc /. 2 is_a_value_on T & loc /. 4 is_a_value_on T & loc /. 6 is_a_value_on T & loc /. 7 is_a_value_on T & loc /. 8 is_a_value_on T & loc /. 9 is_a_value_on T & loc /. 10 is_a_value_on T ) ) & loc is one-to-one holds
<*(Lucas_inv (A,loc,x0,y0,p0,q0,n0)),(Lucas_main_loop (A,loc)),(PP_and ((Equality (A,(loc /. 1),(loc /. 3))),(Lucas_inv (A,loc,x0,y0,p0,q0,n0))))*> is SFHT of (ND (V,A))
let x0, y0, p0, q0 be Integer; :: thesis: for n0 being Nat
for V being non empty set
for loc being b₂ -valued 10 -element FinSequence st A is complex-containing & A is_without_nonatomicND_wrt V & ( for T being TypeSCNominativeData of V,A holds
( loc /. 1 is_a_value_on T & loc /. 2 is_a_value_on T & loc /. 4 is_a_value_on T & loc /. 6 is_a_value_on T & loc /. 7 is_a_value_on T & loc /. 8 is_a_value_on T & loc /. 9 is_a_value_on T & loc /. 10 is_a_value_on T ) ) & loc is one-to-one holds
<*(Lucas_inv (A,loc,x0,y0,p0,q0,n0)),(Lucas_main_loop (A,loc)),(PP_and ((Equality (A,(loc /. 1),(loc /. 3))),(Lucas_inv (A,loc,x0,y0,p0,q0,n0))))*> is SFHT of (ND (V,A))
let n0 be Nat; :: thesis: for V being non empty set
for loc being b₁ -valued 10 -element FinSequence st A is complex-containing & A is_without_nonatomicND_wrt V & ( for T being TypeSCNominativeData of V,A holds
( loc /. 1 is_a_value_on T & loc /. 2 is_a_value_on T & loc /. 4 is_a_value_on T & loc /. 6 is_a_value_on T & loc /. 7 is_a_value_on T & loc /. 8 is_a_value_on T & loc /. 9 is_a_value_on T & loc /. 10 is_a_value_on T ) ) & loc is one-to-one holds
<*(Lucas_inv (A,loc,x0,y0,p0,q0,n0)),(Lucas_main_loop (A,loc)),(PP_and ((Equality (A,(loc /. 1),(loc /. 3))),(Lucas_inv (A,loc,x0,y0,p0,q0,n0))))*> is SFHT of (ND (V,A))
let V be non empty set ; :: thesis: for loc being V -valued 10 -element FinSequence st A is complex-containing & A is_without_nonatomicND_wrt V & ( for T being TypeSCNominativeData of V,A holds
( loc /. 1 is_a_value_on T & loc /. 2 is_a_value_on T & loc /. 4 is_a_value_on T & loc /. 6 is_a_value_on T & loc /. 7 is_a_value_on T & loc /. 8 is_a_value_on T & loc /. 9 is_a_value_on T & loc /. 10 is_a_value_on T ) ) & loc is one-to-one holds
<*(Lucas_inv (A,loc,x0,y0,p0,q0,n0)),(Lucas_main_loop (A,loc)),(PP_and ((Equality (A,(loc /. 1),(loc /. 3))),(Lucas_inv (A,loc,x0,y0,p0,q0,n0))))*> is SFHT of (ND (V,A))
let loc be V -valued 10 -element FinSequence; :: thesis: ( A is complex-containing & A is_without_nonatomicND_wrt V & ( for T being TypeSCNominativeData of V,A holds
( loc /. 1 is_a_value_on T & loc /. 2 is_a_value_on T & loc /. 4 is_a_value_on T & loc /. 6 is_a_value_on T & loc /. 7 is_a_value_on T & loc /. 8 is_a_value_on T & loc /. 9 is_a_value_on T & loc /. 10 is_a_value_on T ) ) & loc is one-to-one implies <*(Lucas_inv (A,loc,x0,y0,p0,q0,n0)),(Lucas_main_loop (A,loc)),(PP_and ((Equality (A,(loc /. 1),(loc /. 3))),(Lucas_inv (A,loc,x0,y0,p0,q0,n0))))*> is SFHT of (ND (V,A)) )
set i = loc /. 1;
set j = loc /. 2;
set n = loc /. 3;
set s = loc /. 4;
set b = loc /. 5;
set c = loc /. 6;
set p = loc /. 7;
set q = loc /. 8;
set ps = loc /. 9;
set qc = loc /. 10;
set D = ND (V,A);
set inv = Lucas_inv (A,loc,x0,y0,p0,q0,n0);
set B = Lucas_loop_body (A,loc);
set E = Equality (A,(loc /. 1),(loc /. 3));
set N = PP_inversion (Lucas_inv (A,loc,x0,y0,p0,q0,n0));
assume ( A is complex-containing & A is_without_nonatomicND_wrt V & ( for T being TypeSCNominativeData of V,A holds
( loc /. 1 is_a_value_on T & loc /. 2 is_a_value_on T & loc /. 4 is_a_value_on T & loc /. 6 is_a_value_on T & loc /. 7 is_a_value_on T & loc /. 8 is_a_value_on T & loc /. 9 is_a_value_on T & loc /. 10 is_a_value_on T ) ) & loc is one-to-one ) ; :: thesis: <*(Lucas_inv (A,loc,x0,y0,p0,q0,n0)),(Lucas_main_loop (A,loc)),(PP_and ((Equality (A,(loc /. 1),(loc /. 3))),(Lucas_inv (A,loc,x0,y0,p0,q0,n0))))*> is SFHT of (ND (V,A))
then <*(Lucas_inv (A,loc,x0,y0,p0,q0,n0)),(Lucas_loop_body (A,loc)),(Lucas_inv (A,loc,x0,y0,p0,q0,n0))*> is SFHT of (ND (V,A)) by Th16;
then A1: <*(PP_and ((PP_not (Equality (A,(loc /. 1),(loc /. 3)))),(Lucas_inv (A,loc,x0,y0,p0,q0,n0)))),(Lucas_loop_body (A,loc)),(Lucas_inv (A,loc,x0,y0,p0,q0,n0))*> is SFHT of (ND (V,A)) by NOMIN_3:3, NOMIN_3:15;
<*(PP_inversion (Lucas_inv (A,loc,x0,y0,p0,q0,n0))),(Lucas_loop_body (A,loc)),(Lucas_inv (A,loc,x0,y0,p0,q0,n0))*> is SFHT of (ND (V,A)) by NOMIN_3:19;
then <*(PP_and ((PP_not (Equality (A,(loc /. 1),(loc /. 3)))),(PP_inversion (Lucas_inv (A,loc,x0,y0,p0,q0,n0))))),(Lucas_loop_body (A,loc)),(Lucas_inv (A,loc,x0,y0,p0,q0,n0))*> is SFHT of (ND (V,A)) by NOMIN_3:3, NOMIN_3:15;
hence <*(Lucas_inv (A,loc,x0,y0,p0,q0,n0)),(Lucas_main_loop (A,loc)),(PP_and ((Equality (A,(loc /. 1),(loc /. 3))),(Lucas_inv (A,loc,x0,y0,p0,q0,n0))))*> is SFHT of (ND (V,A)) by A1, NOMIN_3:26; :: thesis: verum