let p1, p2 be FinSequence of PFuncs (D,REAL); :: thesis: ( len p1 = len f & ( for n being Nat st n in dom p1 holds
p1 . n = chi ((f . n),D) ) & len p2 = len f & ( for n being Nat st n in dom p2 holds
p2 . n = chi ((f . n),D) ) implies p1 = p2 )
assume that
A4: len p1 = len f and
A5: for n being Nat st n in dom p1 holds
p1 . n = chi ((f . n),D) and
A6: len p2 = len f and
A7: for n being Nat st n in dom p2 holds
p2 . n = chi ((f . n),D) ; :: thesis: p1 = p2
A8: ( dom p1 = Seg (len p1) & 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 A9: n in dom p1 ; :: thesis: p1 . n = p2 . n
then p1 . n = chi ((f . n),D) by A5;
hence p1 . n = p2 . n by A4, A6, A7, A8, A9; :: thesis: verum

end;

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