let q, r be FinSequence; :: thesis: ( len q = (len p) - (x .. p) & ( for k being Nat st k in dom q holds
q . k = p . (k + (x .. p)) ) & len r = (len p) - (x .. p) & ( for k being Nat st k in dom r holds
r . k = p . (k + (x .. p)) ) implies q = r )
assume that
A4: len q = (len p) - (x .. p) and
A5: for k being Nat st k in dom q holds
q . k = p . (k + (x .. p)) ; :: thesis: ( not len r = (len p) - (x .. p) or ex k being Nat st
( k in dom r & not r . k = p . (k + (x .. p)) ) or q = r )
assume that
A6: len r = (len p) - (x .. p) and
A7: for k being Nat st k in dom r holds
r . k = p . (k + (x .. p)) ; :: thesis: q = r

now

let k be Nat; :: thesis: ( k in dom q implies q . k = r . k )
assume A8: k in dom q ; :: thesis: q . k = r . k
( dom q = Seg (len q) & dom r = Seg (len r) ) by FINSEQ_1:def 3;
then ( q . k = p . (k + (x .. p)) & r . k = p . (k + (x .. p)) ) by A4, A5, A6, A7, A8;
hence q . k = r . k ; :: thesis: verum

end;

hence q = r by A4, A6, FINSEQ_2:10; :: thesis: verum