let R be non empty ZeroStr ; :: thesis: for p, q being AlgSequence of R st len p = len q & ( for k being Nat st k < len p holds
p . k = q . k ) holds
p = q
let p, q be AlgSequence of R; :: thesis: ( len p = len q & ( for k being Nat st k < len p holds
p . k = q . k ) implies p = q )
assume that
A1: len p = len q and
A2: for k being Nat st k < len p holds
p . k = q . k ; :: thesis: p = q
A3: for x being object st x in NAT holds
p . x = q . x

proof

let x be object ; :: thesis: ( x in NAT implies p . x = q . x )
assume x in NAT ; :: thesis: p . x = q . x
then reconsider k = x as Element of NAT ;
( k >= len p implies p . k = q . k )

proof

assume A4: k >= len p ; :: thesis: p . k = q . k
then p . k = 0. R by Th1;
hence p . k = q . k by A1, A4, Th1; :: thesis: verum

end;

hence p . x = q . x by A2; :: thesis: verum

end;

( dom p = NAT & dom q = NAT ) by FUNCT_2:def 1;
hence p = q by A3, FUNCT_1:2; :: thesis: verum