let X be set ; :: thesis: for A1, A2, A3 being SetSequence of X st ( for n being Nat holds A3 . n = (A1 . n) \/ (A2 . n) ) holds

for n being Nat holds (superior_setsequence A3) . n = ((superior_setsequence A1) . n) \/ ((superior_setsequence A2) . n)

let A1, A2, A3 be SetSequence of X; :: thesis: ( ( for n being Nat holds A3 . n = (A1 . n) \/ (A2 . n) ) implies for n being Nat holds (superior_setsequence A3) . n = ((superior_setsequence A1) . n) \/ ((superior_setsequence A2) . n) )

assume A1: for n being Nat holds A3 . n = (A1 . n) \/ (A2 . n) ; :: thesis: for n being Nat holds (superior_setsequence A3) . n = ((superior_setsequence A1) . n) \/ ((superior_setsequence A2) . n)

let n be Nat; :: thesis: (superior_setsequence A3) . n = ((superior_setsequence A1) . n) \/ ((superior_setsequence A2) . n)

reconsider X3 = superior_setsequence A3 as SetSequence of X ;

reconsider X2 = superior_setsequence A2 as SetSequence of X ;

set B = A1;

reconsider X1 = superior_setsequence A1 as SetSequence of X ;

A2: (X1 . n) \/ (X2 . n) c= X3 . n

for n being Nat holds (superior_setsequence A3) . n = ((superior_setsequence A1) . n) \/ ((superior_setsequence A2) . n)

let A1, A2, A3 be SetSequence of X; :: thesis: ( ( for n being Nat holds A3 . n = (A1 . n) \/ (A2 . n) ) implies for n being Nat holds (superior_setsequence A3) . n = ((superior_setsequence A1) . n) \/ ((superior_setsequence A2) . n) )

assume A1: for n being Nat holds A3 . n = (A1 . n) \/ (A2 . n) ; :: thesis: for n being Nat holds (superior_setsequence A3) . n = ((superior_setsequence A1) . n) \/ ((superior_setsequence A2) . n)

let n be Nat; :: thesis: (superior_setsequence A3) . n = ((superior_setsequence A1) . n) \/ ((superior_setsequence A2) . n)

reconsider X3 = superior_setsequence A3 as SetSequence of X ;

reconsider X2 = superior_setsequence A2 as SetSequence of X ;

set B = A1;

reconsider X1 = superior_setsequence A1 as SetSequence of X ;

A2: (X1 . n) \/ (X2 . n) c= X3 . n

proof

X3 . n c= (X1 . n) \/ (X2 . n)
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (X1 . n) \/ (X2 . n) or x in X3 . n )

assume A3: x in (X1 . n) \/ (X2 . n) ; :: thesis: x in X3 . n

end;assume A3: x in (X1 . n) \/ (X2 . n) ; :: thesis: x in X3 . n

per cases
( x in X1 . n or x in X2 . n )
by A3, XBOOLE_0:def 3;

end;

proof

hence
(superior_setsequence A3) . n = ((superior_setsequence A1) . n) \/ ((superior_setsequence A2) . n)
by A2, XBOOLE_0:def 10; :: thesis: verum
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X3 . n or x in (X1 . n) \/ (X2 . n) )

assume x in X3 . n ; :: thesis: x in (X1 . n) \/ (X2 . n)

then consider k being Nat such that

A6: x in A3 . (n + k) by Th20;

A7: x in (A1 . (n + k)) \/ (A2 . (n + k)) by A1, A6;

hence x in (X1 . n) \/ (X2 . n) by XBOOLE_0:def 3; :: thesis: verum

end;assume x in X3 . n ; :: thesis: x in (X1 . n) \/ (X2 . n)

then consider k being Nat such that

A6: x in A3 . (n + k) by Th20;

A7: x in (A1 . (n + k)) \/ (A2 . (n + k)) by A1, A6;

hence x in (X1 . n) \/ (X2 . n) by XBOOLE_0:def 3; :: thesis: verum