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

proof

A01: S₁[ 0 ] by Th19;
A02: for k being Element of NAT st S₁[k] holds
S₁[k + 1] by A00, Th40;
thus for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A01, A02); :: thesis: verum

end;

hence (inferior_setsequence B) . n = Intersection B ; :: thesis: verum