let n be Element of NAT ; :: thesis: for X being set
for B being SetSequence of X st B is non-descending holds
(superior_setsequence B) . n = Union B
let X be set ; :: thesis: for B being SetSequence of X st B is non-descending holds
(superior_setsequence B) . n = Union B
let B be SetSequence of X; :: thesis: ( B is non-descending implies (superior_setsequence B) . n = Union B )
assume A00: B is non-descending ; :: thesis: (superior_setsequence B) . n = Union B
defpred S₁[ Nat] means (superior_setsequence B) . $1 = Union B;
for n being Element of NAT holds (superior_setsequence B) . n = Union B

proof

A01: S₁[ 0 ] by Th20;
A02: for k being Element of NAT st S₁[k] holds
S₁[k + 1] by A00, Th30;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A01, A02); :: thesis: verum

end;

hence (superior_setsequence B) . n = Union B ; :: thesis: verum