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 c= ((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 c= ((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 c= ((superior_setsequence A1) . n) /\ ((superior_setsequence A2) . n)

let n be Nat; :: thesis: (superior_setsequence A3) . n c= ((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 ;

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

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

let n be Nat; :: thesis: (superior_setsequence A3) . n c= ((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 ;

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

proof

hence
(superior_setsequence A3) . n c= ((superior_setsequence A1) . n) /\ ((superior_setsequence A2) . n)
; :: 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

A2: x in A3 . (n + k) by Th20;

A3: A3 . (n + k) = (A1 . (n + k)) /\ (A2 . (n + k)) by A1;

then x in A2 . (n + k) by A2, XBOOLE_0:def 4;

then A4: x in X2 . n by Th20;

x in A1 . (n + k) by A2, A3, XBOOLE_0:def 4;

then x in X1 . n by Th20;

hence x in (X1 . n) /\ (X2 . n) by A4, XBOOLE_0:def 4; :: thesis: verum

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

then consider k being Nat such that

A2: x in A3 . (n + k) by Th20;

A3: A3 . (n + k) = (A1 . (n + k)) /\ (A2 . (n + k)) by A1;

then x in A2 . (n + k) by A2, XBOOLE_0:def 4;

then A4: x in X2 . n by Th20;

x in A1 . (n + k) by A2, A3, XBOOLE_0:def 4;

then x in X1 . n by Th20;

hence x in (X1 . n) /\ (X2 . n) by A4, XBOOLE_0:def 4; :: thesis: verum