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)

proof

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;

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