let n be Nat; :: thesis: for X being set
for A being Subset of X
for A1 being SetSequence of X holds (inferior_setsequence (A (\/) A1)) . n = A \/ ((inferior_setsequence A1) . n)
let X be set ; :: thesis: for A being Subset of X
for A1 being SetSequence of X holds (inferior_setsequence (A (\/) A1)) . n = A \/ ((inferior_setsequence A1) . n)
let A be Subset of X; :: thesis: for A1 being SetSequence of X holds (inferior_setsequence (A (\/) A1)) . n = A \/ ((inferior_setsequence A1) . n)
let A1 be SetSequence of X; :: thesis: (inferior_setsequence (A (\/) A1)) . n = A \/ ((inferior_setsequence A1) . n)
(inferior_setsequence (A (\/) A1)) . n = Intersection ((A (\/) A1) ^\ n) by Th1
.= Intersection (A (\/) (A1 ^\ n)) by Th17
.= A \/ (Intersection (A1 ^\ n)) by Th34
.= A \/ ((inferior_setsequence A1) . n) by Th1 ;
hence (inferior_setsequence (A (\/) A1)) . n = A \/ ((inferior_setsequence A1) . n) ; :: thesis: verum