let S1, S2 be SetSequence of X; :: thesis: ( S1 . 0 = A1 . 0 & ( for n being Nat holds S1 . (n + 1) = (A1 . (n + 1)) \ ((Partial_Union A1) . n) ) & S2 . 0 = A1 . 0 & ( for n being Nat holds S2 . (n + 1) = (A1 . (n + 1)) \ ((Partial_Union A1) . n) ) implies S1 = S2 )
assume that
A3: S1 . 0 = A1 . 0 and
A4: for n being Nat holds S1 . (n + 1) = (A1 . (n + 1)) \ ((Partial_Union A1) . n) and
A5: S2 . 0 = A1 . 0 and
A6: for n being Nat holds S2 . (n + 1) = (A1 . (n + 1)) \ ((Partial_Union A1) . n) ; :: thesis: S1 = S2
defpred S₁[ object ] means S1 . $1 = S2 . $1;
for n being object st n in NAT holds
S₁[n]

proof

let n be object ; :: thesis: ( n in NAT implies S₁[n] )
assume n in NAT ; :: thesis: S₁[n]
then reconsider n = n as Element of NAT ;
A7: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume S1 . k = S2 . k ; :: thesis: S₁[k + 1]
thus S1 . (k + 1) = (A1 . (k + 1)) \ ((Partial_Union A1) . k) by A4
.= S2 . (k + 1) by A6 ; :: thesis: verum

end;

A8: S₁[ 0 ] by A3, A5;
for k being Nat holds S₁[k] from NAT_1:sch 2(A8, A7);
then S1 . n = S2 . n ;
hence S₁[n] ; :: thesis: verum

end;

hence S1 = S2 ; :: thesis: verum