let x, y be Element of INT ; :: thesis: 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 Element of 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 Element of 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; :: thesis: ( 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 Element of 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 Element of 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 P1: ( 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 Element of 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) ) ) ) ; :: thesis: ( 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 Element of 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 P2: ( 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 Element of 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) ) ) ) ; :: thesis: ( 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 );
P0: S₁[ 0 ] by P1, P2;
PN: for n being Element of NAT st S₁[n] holds
S₁[n + 1]

proof

let n be Element of NAT ; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume PN1: S₁[n] ; :: thesis: S₁[n + 1]
X1: q1 . (n + 1) = (g2 . n) div (w2 . n) by P1, PN1
.= q2 . (n + 1) by P2 ;
X2: t1 . (n + 1) = (g2 . n) mod (w2 . n) by P1, PN1
.= t2 . (n + 1) by P2 ;
X3: a1 . (n + 1) = u2 . n by P1, PN1
.= a2 . (n + 1) by P2 ;
X4: b1 . (n + 1) = v2 . n by PN1, P1
.= b2 . (n + 1) by P2 ;
X5: g1 . (n + 1) = w2 . n by PN1, P1
.= g2 . (n + 1) by P2 ;
X6: u1 . (n + 1) = (a2 . n) - ((q1 . (n + 1)) * (u2 . n)) by P1, PN1
.= u2 . (n + 1) by P2, X1 ;
X7: v1 . (n + 1) = (b2 . n) - ((q1 . (n + 1)) * (v2 . n)) by PN1, P1
.= v2 . (n + 1) by P2, X1 ;
w1 . (n + 1) = t2 . (n + 1) by X2, P1
.= w2 . (n + 1) by P2 ;
hence S₁[n + 1] by X1, X2, X3, X4, X5, X6, X7; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(P0, PN);
hence ( g1 = g2 & w1 = w2 & q1 = q2 & t1 = t2 & a1 = a2 & b1 = b2 & v1 = v2 & u1 = u2 ) by FUNCT_2:def 8; :: thesis: verum