let S1, S2 be SetSequence of REAL; :: thesis: ( S1 . 0 = Union (GoCross_Partial_Union (pm,0)) & ( for n being Nat holds S1 . (n + 1) = (S1 . n) \/ (Union (GoCross_Partial_Union (pm,(n + 1)))) ) & S2 . 0 = Union (GoCross_Partial_Union (pm,0)) & ( for n being Nat holds S2 . (n + 1) = (S2 . n) \/ (Union (GoCross_Partial_Union (pm,(n + 1)))) ) implies S1 = S2 )
assume that
A4: S1 . 0 = Union (GoCross_Partial_Union (pm,0)) and
A5: for n being Nat holds S1 . (n + 1) = (S1 . n) \/ (Union (GoCross_Partial_Union (pm,(n + 1)))) and
A6: S2 . 0 = Union (GoCross_Partial_Union (pm,0)) and
A7: for n being Nat holds S2 . (n + 1) = (S2 . n) \/ (Union (GoCross_Partial_Union (pm,(n + 1)))) ; :: 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 Nat ;
A8: for k2 being Nat st S₁[k2] holds
S₁[k2 + 1]

proof

let k2 be Nat; :: thesis: ( S₁[k2] implies S₁[k2 + 1] )
assume S1 . k2 = S2 . k2 ; :: thesis: S₁[k2 + 1]
hence S1 . (k2 + 1) = (S2 . k2) \/ (Union (GoCross_Partial_Union (pm,(k2 + 1)))) by A5
.= S2 . (k2 + 1) by A7 ;
:: thesis: verum

end;

A9: S₁[ 0 ] by A4, A6;
for k being Nat holds S₁[k] from NAT_1:sch 2(A9, A8);
then S1 . n = S2 . n ;
hence S₁[n] ; :: thesis: verum

end;

hence S1 = S2 ; :: thesis: verum