let a, b be SetSequence; :: thesis: ( ( for n being Nat holds a . n = { f where f is Element of Funcs ((Seg n),(Seg (n + 1))) : @ f is increasing } ) & ( for n being Nat holds b . n = { f where f is Element of Funcs ((Seg n),(Seg (n + 1))) : @ f is increasing } ) implies a = b )
assume that
A2: for n being Nat holds a . n = { f where f is Element of Funcs ((Seg n),(Seg (n + 1))) : @ f is increasing } and
A3: for n being Nat holds b . n = { f where f is Element of Funcs ((Seg n),(Seg (n + 1))) : @ f is increasing } ; :: 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 = { f where f is Element of Funcs ((Seg n1),(Seg (n1 + 1))) : @ f is increasing } by A2;
hence b . n c= a . n by A3; :: thesis: verum

end;

then A4: b c= a by PBOOLE:def 2;

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 = { f where f is Element of Funcs ((Seg n1),(Seg (n1 + 1))) : @ f is increasing } by A2;
hence a . n c= b . n by A3; :: thesis: verum

end;

then a c= b by PBOOLE:def 2;
hence a = b by A4, PBOOLE:def 10; :: thesis: verum