let s1, s2 be State of S; :: thesis: ( ex f being sequence of (product (the_Values_of S)) st
( s1 = f . k & f . 0 = s & ( for i being Nat holds f . (i + 1) = Following (p,(f . i)) ) ) & ex f being sequence of (product (the_Values_of S)) st
( s2 = f . k & f . 0 = s & ( for i being Nat holds f . (i + 1) = Following (p,(f . i)) ) ) implies s1 = s2 )
given f1 being sequence of (product (the_Values_of S)) such that A3: s1 = f1 . k and
A4: f1 . 0 = s and
A5: for i being Nat holds f1 . (i + 1) = Following (p,(f1 . i)) ; :: thesis: ( for f being sequence of (product (the_Values_of S)) holds
( not s2 = f . k or not f . 0 = s or ex i being Nat st not f . (i + 1) = Following (p,(f . i)) ) or s1 = s2 )
given f2 being sequence of (product (the_Values_of S)) such that A6: s2 = f2 . k and
A7: f2 . 0 = s and
A8: for i being Nat holds f2 . (i + 1) = Following (p,(f2 . i)) ; :: thesis: s1 = s2
reconsider s = s as Element of product (the_Values_of S) by CARD_3:107;
A9: f1 . 0 = s by A4;
A10: for i being Nat holds f1 . (i + 1) = H₁(i,f1 . i) by A5;
A11: f2 . 0 = s by A7;
A12: for i being Nat holds f2 . (i + 1) = H₁(i,f2 . i) by A8;
f1 = f2 from NAT_1:sch 16(A9, A10, A11, A12);
hence s1 = s2 by A3, A6; :: thesis: verum