let U be non empty set ; :: thesis: for A, B being non empty IntervalSet of U holds A _/\_ B = Inter (((A ``1) /\ (B ``1)),((A ``2) /\ (B ``2)))

let A, B be non empty IntervalSet of U; :: thesis: A _/\_ B = Inter (((A ``1) /\ (B ``1)),((A ``2) /\ (B ``2)))

A1: A _/\_ B c= Inter (((A ``1) /\ (B ``1)),((A ``2) /\ (B ``2)))

let A, B be non empty IntervalSet of U; :: thesis: A _/\_ B = Inter (((A ``1) /\ (B ``1)),((A ``2) /\ (B ``2)))

A1: A _/\_ B c= Inter (((A ``1) /\ (B ``1)),((A ``2) /\ (B ``2)))

proof

Inter (((A ``1) /\ (B ``1)),((A ``2) /\ (B ``2))) c= A _/\_ B
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in A _/\_ B or x in Inter (((A ``1) /\ (B ``1)),((A ``2) /\ (B ``2))) )

reconsider xx = x as set by TARSKI:1;

assume x in A _/\_ B ; :: thesis: x in Inter (((A ``1) /\ (B ``1)),((A ``2) /\ (B ``2)))

then consider X, Y being set such that

A2: ( X in A & Y in B & x = X /\ Y ) by SETFAM_1:def 5;

( A ``1 c= X & B ``1 c= Y & X c= A ``2 & Y c= B ``2 ) by A2, Th14;

then ( (A ``1) /\ (B ``1) c= xx & xx c= (A ``2) /\ (B ``2) ) by A2, XBOOLE_1:27;

hence x in Inter (((A ``1) /\ (B ``1)),((A ``2) /\ (B ``2))) by Th1; :: thesis: verum

end;reconsider xx = x as set by TARSKI:1;

assume x in A _/\_ B ; :: thesis: x in Inter (((A ``1) /\ (B ``1)),((A ``2) /\ (B ``2)))

then consider X, Y being set such that

A2: ( X in A & Y in B & x = X /\ Y ) by SETFAM_1:def 5;

( A ``1 c= X & B ``1 c= Y & X c= A ``2 & Y c= B ``2 ) by A2, Th14;

then ( (A ``1) /\ (B ``1) c= xx & xx c= (A ``2) /\ (B ``2) ) by A2, XBOOLE_1:27;

hence x in Inter (((A ``1) /\ (B ``1)),((A ``2) /\ (B ``2))) by Th1; :: thesis: verum

proof

hence
A _/\_ B = Inter (((A ``1) /\ (B ``1)),((A ``2) /\ (B ``2)))
by A1; :: thesis: verum
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Inter (((A ``1) /\ (B ``1)),((A ``2) /\ (B ``2))) or x in A _/\_ B )

assume x in Inter (((A ``1) /\ (B ``1)),((A ``2) /\ (B ``2))) ; :: thesis: x in A _/\_ B

then consider Z being Element of bool U such that

A3: ( x = Z & (A ``1) /\ (B ``1) c= Z & Z c= (A ``2) /\ (B ``2) ) ;

A ``1 c= (Z \/ (A ``1)) /\ (A ``2)

then A7: (Z \/ (A ``1)) /\ (A ``2) in A by Th14;

B ``1 c= (Z \/ (B ``1)) /\ (B ``2)

then A10: (Z \/ (B ``1)) /\ (B ``2) in B by Th14;

((Z \/ (A ``1)) /\ (A ``2)) /\ ((Z \/ (B ``1)) /\ (B ``2)) = (((A ``2) /\ (Z \/ (A ``1))) /\ (Z \/ (B ``1))) /\ (B ``2) by XBOOLE_1:16

.= ((A ``2) /\ ((Z \/ (A ``1)) /\ (Z \/ (B ``1)))) /\ (B ``2) by XBOOLE_1:16

.= ((A ``2) /\ (Z \/ ((A ``1) /\ (B ``1)))) /\ (B ``2) by XBOOLE_1:24

.= ((A ``2) /\ Z) /\ (B ``2) by A3, XBOOLE_1:12

.= Z /\ ((A ``2) /\ (B ``2)) by XBOOLE_1:16

.= Z by A3, XBOOLE_1:28 ;

hence x in A _/\_ B by A3, A7, A10, SETFAM_1:def 5; :: thesis: verum

end;assume x in Inter (((A ``1) /\ (B ``1)),((A ``2) /\ (B ``2))) ; :: thesis: x in A _/\_ B

then consider Z being Element of bool U such that

A3: ( x = Z & (A ``1) /\ (B ``1) c= Z & Z c= (A ``2) /\ (B ``2) ) ;

A ``1 c= (Z \/ (A ``1)) /\ (A ``2)

proof

then
( A ``1 c= (Z \/ (A ``1)) /\ (A ``2) & (Z \/ (A ``1)) /\ (A ``2) c= A ``2 )
by XBOOLE_1:17;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in A ``1 or x in (Z \/ (A ``1)) /\ (A ``2) )

assume A4: x in A ``1 ; :: thesis: x in (Z \/ (A ``1)) /\ (A ``2)

assume A5: not x in (Z \/ (A ``1)) /\ (A ``2) ; :: thesis: contradiction

end;assume A4: x in A ``1 ; :: thesis: x in (Z \/ (A ``1)) /\ (A ``2)

assume A5: not x in (Z \/ (A ``1)) /\ (A ``2) ; :: thesis: contradiction

then A7: (Z \/ (A ``1)) /\ (A ``2) in A by Th14;

B ``1 c= (Z \/ (B ``1)) /\ (B ``2)

proof

then
( B ``1 c= (Z \/ (B ``1)) /\ (B ``2) & (Z \/ (B ``1)) /\ (B ``2) c= B ``2 )
by XBOOLE_1:17;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in B ``1 or x in (Z \/ (B ``1)) /\ (B ``2) )

assume A8: x in B ``1 ; :: thesis: x in (Z \/ (B ``1)) /\ (B ``2)

then A9: x in Z \/ (B ``1) by XBOOLE_0:def 3;

B ``1 c= B ``2 by Th16;

hence x in (Z \/ (B ``1)) /\ (B ``2) by A8, A9, XBOOLE_0:def 4; :: thesis: verum

end;assume A8: x in B ``1 ; :: thesis: x in (Z \/ (B ``1)) /\ (B ``2)

then A9: x in Z \/ (B ``1) by XBOOLE_0:def 3;

B ``1 c= B ``2 by Th16;

hence x in (Z \/ (B ``1)) /\ (B ``2) by A8, A9, XBOOLE_0:def 4; :: thesis: verum

then A10: (Z \/ (B ``1)) /\ (B ``2) in B by Th14;

((Z \/ (A ``1)) /\ (A ``2)) /\ ((Z \/ (B ``1)) /\ (B ``2)) = (((A ``2) /\ (Z \/ (A ``1))) /\ (Z \/ (B ``1))) /\ (B ``2) by XBOOLE_1:16

.= ((A ``2) /\ ((Z \/ (A ``1)) /\ (Z \/ (B ``1)))) /\ (B ``2) by XBOOLE_1:16

.= ((A ``2) /\ (Z \/ ((A ``1) /\ (B ``1)))) /\ (B ``2) by XBOOLE_1:24

.= ((A ``2) /\ Z) /\ (B ``2) by A3, XBOOLE_1:12

.= Z /\ ((A ``2) /\ (B ``2)) by XBOOLE_1:16

.= Z by A3, XBOOLE_1:28 ;

hence x in A _/\_ B by A3, A7, A10, SETFAM_1:def 5; :: thesis: verum