let p1, p2 be PFuncFinSequence of A; :: thesis: ( dom p1 = dom the charact of U0 & ( for n being set
for o being operation of U0 st n in dom p1 & o = the charact of U0 . n holds
p1 . n = o /. A ) & dom p2 = dom the charact of U0 & ( for n being set
for o being operation of U0 st n in dom p2 & o = the charact of U0 . n holds
p2 . n = o /. A ) implies p1 = p2 )

assume that
A4: dom p1 = dom the charact of U0 and
A5: for n being set
for o being operation of U0 st n in dom p1 & o = the charact of U0 . n holds
p1 . n = o /. A and
A6: dom p2 = dom the charact of U0 and
A7: for n being set
for o being operation of U0 st n in dom p2 & o = the charact of U0 . n holds
p2 . n = o /. A ; :: thesis: p1 = p2
for n being Nat st n in dom the charact of U0 holds
p1 . n = p2 . n
proof
let n be Nat; :: thesis: ( n in dom the charact of U0 implies p1 . n = p2 . n )
assume A8: n in dom the charact of U0 ; :: thesis: p1 . n = p2 . n
then reconsider k = the charact of U0 . n as operation of U0 by FUNCT_1:def 3;
p1 . n = k /. A by A4, A5, A8;
hence p1 . n = p2 . n by A6, A7, A8; :: thesis: verum
end;
hence p1 = p2 by ; :: thesis: verum