let a, b be SetSequence; :: thesis: ( ( for n being Nat holds a . n = Funcs ((X . (n + 1)),(X . n)) ) & ( for n being Nat holds b . n = Funcs ((X . (n + 1)),(X . n)) ) implies a = b )
assume that
A2: for n being Nat holds a . n = Funcs ((X . (n + 1)),(X . n)) and
A3: for n being Nat holds b . n = Funcs ((X . (n + 1)),(X . n)) ; :: thesis: a = b

now

let n be set ; :: thesis: ( n in NAT implies b . n c= a . n )
assume n in NAT ; :: thesis: b . n c= a . n
then reconsider n1 = n as Element of NAT ;
a . n1 = Funcs ((X . (n1 + 1)),(X . n1)) by A2;
hence b . n c= a . n by A3; :: thesis: verum

end;

then A4: b c= a by PBOOLE:def 5;

now

let n be set ; :: thesis: ( n in NAT implies a . n c= b . n )
assume n in NAT ; :: thesis: a . n c= b . n
then reconsider n1 = n as Element of NAT ;
a . n1 = Funcs ((X . (n1 + 1)),(X . n1)) by A2;
hence a . n c= b . n by A3; :: thesis: verum

end;

then a c= b by PBOOLE:def 5;
hence a = b by A4, PBOOLE:def 13; :: thesis: verum