let p1, p2 be FinSequence of REAL ; :: thesis: ( len p1 = width m & ( for j being Nat st j in Seg (width m) holds
p1 . j = Sum (Col m,j) ) & len p2 = width m & ( for j being Nat st j in Seg (width m) holds
p2 . j = Sum (Col m,j) ) implies p1 = p2 )
assume that
A3: ( len p1 = width m & ( for i being Nat st i in Seg (width m) holds
p1 . i = Sum (Col m,i) ) ) and
A4: ( len p2 = width m & ( for i being Nat st i in Seg (width m) holds
p2 . i = Sum (Col m,i) ) ) ; :: thesis: p1 = p2
X: dom p1 = Seg (width m) by A3, FINSEQ_1:def 3;
for j being Nat st j in dom p1 holds
p1 . j = p2 . j

proof

let j be Nat; :: thesis: ( j in dom p1 implies p1 . j = p2 . j )
assume j in dom p1 ; :: thesis: p1 . j = p2 . j
then ( p1 . j = Sum (Col m,j) & p2 . j = Sum (Col m,j) ) by A3, A4, X;
hence p1 . j = p2 . j ; :: thesis: verum

end;

hence p1 = p2 by A3, A4, FINSEQ_2:10; :: thesis: verum