let x, y be Integer; :: 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 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; :: 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 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) ) ) ) ; :: 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 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) ) ) ) ; :: 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 );
A3: S₁[ 0 ] by A1, A2;
A4: for n being Nat st S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume A5: S₁[n] ; :: thesis: 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; :: thesis: 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 ) ; :: thesis: verum