let p1, p2 be FinSequence of REAL ; :: thesis: ( len p1 = len f & ( for n being Element of NAT st n in dom p1 holds
p1 . n = (f . n) . d ) & len p2 = len f & ( for n being Element of NAT st n in dom p2 holds
p2 . n = (f . n) . d ) implies p1 = p2 )
assume that
A4: len p1 = len f and
A5: for n being Element of NAT st n in dom p1 holds
p1 . n = (f . n) . d and
A6: len p2 = len f and
A7: for n being Element of NAT st n in dom p2 holds
p2 . n = (f . n) . d ; :: thesis: p1 = p2
A8: dom p1 = Seg (len p1) by FINSEQ_1:def 3;
A9: dom p2 = Seg (len p2) by FINSEQ_1:def 3;

now :: thesis: for n being Nat st n in dom p1 holds
p1 . n = p2 . n

let n be Nat; :: thesis: ( n in dom p1 implies p1 . n = p2 . n )
assume A10: n in dom p1 ; :: thesis: p1 . n = p2 . n
then p1 . n = (f . n) . d by A5;
hence p1 . n = p2 . n by A4, A6, A7, A8, A9, A10; :: thesis: verum

end;

hence p1 = p2 by A4, A6, FINSEQ_2:9; :: thesis: verum