let d be non zero Nat; :: thesis: for l, r, l9, r9 being Element of REAL d st ( for i being Element of Seg d holds l . i <= r . i ) & ( for i being Element of Seg d holds r9 . i < l9 . i ) holds
( cell (l,r) c= cell (l9,r9) iff ex i being Element of Seg d st
( r . i <= r9 . i or l9 . i <= l . i ) )
let l, r, l9, r9 be Element of REAL d; :: thesis: ( ( for i being Element of Seg d holds l . i <= r . i ) & ( for i being Element of Seg d holds r9 . i < l9 . i ) implies ( cell (l,r) c= cell (l9,r9) iff ex i being Element of Seg d st
( r . i <= r9 . i or l9 . i <= l . i ) ) )
assume A1: for i being Element of Seg d holds l . i <= r . i ; :: thesis: ( ex i being Element of Seg d st not r9 . i < l9 . i or ( cell (l,r) c= cell (l9,r9) iff ex i being Element of Seg d st
( r . i <= r9 . i or l9 . i <= l . i ) ) )
assume A2: for i being Element of Seg d holds r9 . i < l9 . i ; :: thesis: ( cell (l,r) c= cell (l9,r9) iff ex i being Element of Seg d st
( r . i <= r9 . i or l9 . i <= l . i ) )
thus ( cell (l,r) c= cell (l9,r9) implies ex i being Element of Seg d st
( r . i <= r9 . i or l9 . i <= l . i ) ) :: thesis: ( ex i being Element of Seg d st
( r . i <= r9 . i or l9 . i <= l . i ) implies cell (l,r) c= cell (l9,r9) )

proof

assume A3: cell (l,r) c= cell (l9,r9) ; :: thesis: ex i being Element of Seg d st
( r . i <= r9 . i or l9 . i <= l . i )
assume A4: for i being Element of Seg d holds
( r9 . i < r . i & l . i < l9 . i ) ; :: thesis: contradiction
defpred S₁[ Element of Seg d, Element of REAL ] means ( l . $1 <= $2 & $2 <= r . $1 & r9 . $1 < $2 & $2 < l9 . $1 );
A5: for i being Element of Seg d ex xi being Element of REAL st S₁[i,xi]

proof

let i be Element of Seg d; :: thesis: ex xi being Element of REAL st S₁[i,xi]

per cases ( ( l . i <= r9 . i & l9 . i <= r . i ) or ( r9 . i < l . i & l9 . i <= r . i ) or r . i < l9 . i ) ;

suppose A6: ( l . i <= r9 . i & l9 . i <= r . i ) ; :: thesis: ex xi being Element of REAL st S₁[i,xi]

r9 . i < l9 . i by A2;
then consider xi being Element of REAL such that
A7: r9 . i < xi and
A8: xi < l9 . i by Th1;
take xi ; :: thesis: S₁[i,xi]
thus S₁[i,xi] by A6, A7, A8, XXREAL_0:2; :: thesis: verum

end;

suppose A9: ( r9 . i < l . i & l9 . i <= r . i ) ; :: thesis: ex xi being Element of REAL st S₁[i,xi]

reconsider li = l . i as Element of REAL by XREAL_0:def 1;
take li ; :: thesis: S₁[i,li]
thus S₁[i,li] by A1, A4, A9; :: thesis: verum

end;

suppose A10: r . i < l9 . i ; :: thesis: ex xi being Element of REAL st S₁[i,xi]

reconsider ri = r . i as Element of REAL by XREAL_0:def 1;
take ri ; :: thesis: S₁[i,ri]
thus S₁[i,ri] by A1, A4, A10; :: thesis: verum

end;

end;

end;

consider x being Function of (Seg d),REAL such that
A11: for i being Element of Seg d holds S₁[i,x . i] from FUNCT_2:sch 3(A5);
reconsider x = x as Element of REAL d by Def3;
A12: x in cell (l,r) by A11;
set i0 = the Element of Seg d;
r9 . the Element of Seg d < l9 . the Element of Seg d by A2;
then ex i being Element of Seg d st
( r9 . i < l9 . i & ( x . i <= r9 . i or l9 . i <= x . i ) ) by A3, A12, Th22;
hence contradiction by A11; :: thesis: verum

end;

given i0 being Element of Seg d such that A13: ( r . i0 <= r9 . i0 or l9 . i0 <= l . i0 ) ; :: thesis: cell (l,r) c= cell (l9,r9)
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in cell (l,r) or x in cell (l9,r9) )
assume A14: x in cell (l,r) ; :: thesis: x in cell (l9,r9)
then reconsider x = x as Element of REAL d ;
A15: l . i0 <= x . i0 by A1, A14, Th21;
A16: x . i0 <= r . i0 by A1, A14, Th21;
ex i being Element of Seg d st
( r9 . i < l9 . i & ( x . i <= r9 . i or l9 . i <= x . i ) )

proof

take i0 ; :: thesis: ( r9 . i0 < l9 . i0 & ( x . i0 <= r9 . i0 or l9 . i0 <= x . i0 ) )
thus ( r9 . i0 < l9 . i0 & ( x . i0 <= r9 . i0 or l9 . i0 <= x . i0 ) ) by A2, A13, A15, A16, XXREAL_0:2; :: thesis: verum

end;

hence x in cell (l9,r9) ; :: thesis: verum