let x, y be Integer; for g1, w1, q1, t1, a1, b1, v1, u1, g2, w2, q2, t2, a2, b2, v2, u2 being sequence of INT st a1 . 0 = 1 & b1 . 0 = 0 & g1 . 0 = x & q1 . 0 = 0 & u1 . 0 = 0 & v1 . 0 = 1 & w1 . 0 = y & t1 . 0 = 0 & ( for i being Nat holds
( q1 . (i + 1) = (g1 . i) div (w1 . i) & t1 . (i + 1) = (g1 . i) mod (w1 . i) & a1 . (i + 1) = u1 . i & b1 . (i + 1) = v1 . i & g1 . (i + 1) = w1 . i & u1 . (i + 1) = (a1 . i) - ((q1 . (i + 1)) * (u1 . i)) & v1 . (i + 1) = (b1 . i) - ((q1 . (i + 1)) * (v1 . i)) & w1 . (i + 1) = t1 . (i + 1) ) ) & a2 . 0 = 1 & b2 . 0 = 0 & g2 . 0 = x & q2 . 0 = 0 & u2 . 0 = 0 & v2 . 0 = 1 & w2 . 0 = y & t2 . 0 = 0 & ( for i being Nat holds
( q2 . (i + 1) = (g2 . i) div (w2 . i) & t2 . (i + 1) = (g2 . i) mod (w2 . i) & a2 . (i + 1) = u2 . i & b2 . (i + 1) = v2 . i & g2 . (i + 1) = w2 . i & u2 . (i + 1) = (a2 . i) - ((q2 . (i + 1)) * (u2 . i)) & v2 . (i + 1) = (b2 . i) - ((q2 . (i + 1)) * (v2 . i)) & w2 . (i + 1) = t2 . (i + 1) ) ) holds
( g1 = g2 & w1 = w2 & q1 = q2 & t1 = t2 & a1 = a2 & b1 = b2 & v1 = v2 & u1 = u2 )
let g1, w1, q1, t1, a1, b1, v1, u1, g2, w2, q2, t2, a2, b2, v2, u2 be sequence of INT; ( a1 . 0 = 1 & b1 . 0 = 0 & g1 . 0 = x & q1 . 0 = 0 & u1 . 0 = 0 & v1 . 0 = 1 & w1 . 0 = y & t1 . 0 = 0 & ( for i being Nat holds
( q1 . (i + 1) = (g1 . i) div (w1 . i) & t1 . (i + 1) = (g1 . i) mod (w1 . i) & a1 . (i + 1) = u1 . i & b1 . (i + 1) = v1 . i & g1 . (i + 1) = w1 . i & u1 . (i + 1) = (a1 . i) - ((q1 . (i + 1)) * (u1 . i)) & v1 . (i + 1) = (b1 . i) - ((q1 . (i + 1)) * (v1 . i)) & w1 . (i + 1) = t1 . (i + 1) ) ) & a2 . 0 = 1 & b2 . 0 = 0 & g2 . 0 = x & q2 . 0 = 0 & u2 . 0 = 0 & v2 . 0 = 1 & w2 . 0 = y & t2 . 0 = 0 & ( for i being Nat holds
( q2 . (i + 1) = (g2 . i) div (w2 . i) & t2 . (i + 1) = (g2 . i) mod (w2 . i) & a2 . (i + 1) = u2 . i & b2 . (i + 1) = v2 . i & g2 . (i + 1) = w2 . i & u2 . (i + 1) = (a2 . i) - ((q2 . (i + 1)) * (u2 . i)) & v2 . (i + 1) = (b2 . i) - ((q2 . (i + 1)) * (v2 . i)) & w2 . (i + 1) = t2 . (i + 1) ) ) implies ( g1 = g2 & w1 = w2 & q1 = q2 & t1 = t2 & a1 = a2 & b1 = b2 & v1 = v2 & u1 = u2 ) )
assume A1:
( a1 . 0 = 1 & b1 . 0 = 0 & g1 . 0 = x & q1 . 0 = 0 & u1 . 0 = 0 & v1 . 0 = 1 & w1 . 0 = y & t1 . 0 = 0 & ( for i being Nat holds
( q1 . (i + 1) = (g1 . i) div (w1 . i) & t1 . (i + 1) = (g1 . i) mod (w1 . i) & a1 . (i + 1) = u1 . i & b1 . (i + 1) = v1 . i & g1 . (i + 1) = w1 . i & u1 . (i + 1) = (a1 . i) - ((q1 . (i + 1)) * (u1 . i)) & v1 . (i + 1) = (b1 . i) - ((q1 . (i + 1)) * (v1 . i)) & w1 . (i + 1) = t1 . (i + 1) ) ) )
; ( not a2 . 0 = 1 or not b2 . 0 = 0 or not g2 . 0 = x or not q2 . 0 = 0 or not u2 . 0 = 0 or not v2 . 0 = 1 or not w2 . 0 = y or not t2 . 0 = 0 or ex i being Nat st
( q2 . (i + 1) = (g2 . i) div (w2 . i) & t2 . (i + 1) = (g2 . i) mod (w2 . i) & a2 . (i + 1) = u2 . i & b2 . (i + 1) = v2 . i & g2 . (i + 1) = w2 . i & u2 . (i + 1) = (a2 . i) - ((q2 . (i + 1)) * (u2 . i)) & v2 . (i + 1) = (b2 . i) - ((q2 . (i + 1)) * (v2 . i)) implies not w2 . (i + 1) = t2 . (i + 1) ) or ( g1 = g2 & w1 = w2 & q1 = q2 & t1 = t2 & a1 = a2 & b1 = b2 & v1 = v2 & u1 = u2 ) )
assume A2:
( a2 . 0 = 1 & b2 . 0 = 0 & g2 . 0 = x & q2 . 0 = 0 & u2 . 0 = 0 & v2 . 0 = 1 & w2 . 0 = y & t2 . 0 = 0 & ( for i being Nat holds
( q2 . (i + 1) = (g2 . i) div (w2 . i) & t2 . (i + 1) = (g2 . i) mod (w2 . i) & a2 . (i + 1) = u2 . i & b2 . (i + 1) = v2 . i & g2 . (i + 1) = w2 . i & u2 . (i + 1) = (a2 . i) - ((q2 . (i + 1)) * (u2 . i)) & v2 . (i + 1) = (b2 . i) - ((q2 . (i + 1)) * (v2 . i)) & w2 . (i + 1) = t2 . (i + 1) ) ) )
; ( g1 = g2 & w1 = w2 & q1 = q2 & t1 = t2 & a1 = a2 & b1 = b2 & v1 = v2 & u1 = u2 )
defpred S1[ Nat] means ( g1 . $1 = g2 . $1 & w1 . $1 = w2 . $1 & q1 . $1 = q2 . $1 & t1 . $1 = t2 . $1 & a1 . $1 = a2 . $1 & b1 . $1 = b2 . $1 & v1 . $1 = v2 . $1 & u1 . $1 = u2 . $1 );
A3:
S1[ 0 ]
by A1, A2;
A4:
for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be
Nat;
( S1[n] implies S1[n + 1] )
assume A5:
S1[
n]
;
S1[n + 1]
A6:
q1 . (n + 1) =
(g2 . n) div (w2 . n)
by A1, A5
.=
q2 . (n + 1)
by A2
;
A7:
t1 . (n + 1) =
(g2 . n) mod (w2 . n)
by A1, A5
.=
t2 . (n + 1)
by A2
;
A8:
a1 . (n + 1) =
u2 . n
by A1, A5
.=
a2 . (n + 1)
by A2
;
A9:
b1 . (n + 1) =
v2 . n
by A5, A1
.=
b2 . (n + 1)
by A2
;
A10:
g1 . (n + 1) =
w2 . n
by A5, A1
.=
g2 . (n + 1)
by A2
;
A11:
u1 . (n + 1) =
(a2 . n) - ((q1 . (n + 1)) * (u2 . n))
by A1, A5
.=
u2 . (n + 1)
by A2, A6
;
A12:
v1 . (n + 1) =
(b2 . n) - ((q1 . (n + 1)) * (v2 . n))
by A5, A1
.=
v2 . (n + 1)
by A2, A6
;
w1 . (n + 1) =
t2 . (n + 1)
by A7, A1
.=
w2 . (n + 1)
by A2
;
hence
S1[
n + 1]
by A6, A7, A8, A9, A10, A11, A12;
verum
end;
for n being Nat holds S1[n]
from NAT_1:sch 2(A3, A4);
hence
( g1 = g2 & w1 = w2 & q1 = q2 & t1 = t2 & a1 = a2 & b1 = b2 & v1 = v2 & u1 = u2 )
; verum