
theorem
for S be Subset-Family of REAL
 st S = { I where I is Subset of REAL : I is right_open_interval }
 holds S is with_empty_element semi-diff-closed cap-closed
proof
   let S be Subset-Family of REAL;
   assume A1: S = { I where I is Subset of REAL : I is right_open_interval };
   0 in REAL by NUMBERS:19; then
   [. 0,0 .[ is right_open_interval by NUMBERS:31; then
   {} in S by A1;
   hence S is with_empty_element;
   now let A,B be set;
    assume A2: A in S & B in S; then
    consider I be Subset of REAL such that
A3:  A = I & I is right_open_interval by A1;
    consider J be Subset of REAL such that
A4:  B = J & J is right_open_interval by A1,A2;
    per cases;
    suppose I meets J; then
     consider K,L be Subset of REAL such that
A5:   K is right_open_interval & L is right_open_interval
    & K misses L & I \ J = K \/ L by A3,A4,INTERVAL02;
     set F = <*K,L*>;
     K in S & L in S by A1,A5; then
     {K,L} c= S by ZFMISC_1:32; then
     rng F c= S by FINSEQ_2:127; then
     reconsider F as FinSequence of S by FINSEQ_1:def 4;
     reconsider F as disjoint_valued FinSequence of S by A5,Disjoint2;
     take F;
     rng F = {K,L} by FINSEQ_2:127;
     hence Union F = A \ B by A3,A4,A5,ZFMISC_1:75;
    end;
    suppose A13: I misses J;
     set F = <*I*>;
     {I} c= S by A2,A3,ZFMISC_1:31; then
     dom F = Seg 1 & rng F c= S by FINSEQ_1:38; then
     reconsider F as FinSequence of S by FINSEQ_1:def 4;
     reconsider F as disjoint_valued FinSequence of S by TTT1;
     take F;
     rng F = {I} by FINSEQ_1:38;
     hence Union F = A \ B by A13,A3,A4,XBOOLE_1:83;
    end;
   end;
   hence S is semi-diff-closed;
   now let A,B be set;
    assume B2: A in S & B in S; then
    consider I be Subset of REAL such that
B3:  A = I & I is right_open_interval by A1;
    consider J be Subset of REAL such that
B4:  B = J & J is right_open_interval by A1,B2;
    I /\ J is right_open_interval by B3,B4,INTERVAL03;
    hence A /\ B in S by A1,B3,B4;
   end;
   hence S is cap-closed;
end;
