let a, b, c, d be real number ; :: thesis: ( a <= b & c <= d implies (LSeg |[a,c]|,|[a,d]|) /\ (LSeg |[a,c]|,|[b,c]|) = {|[a,c]|} )
assume that
A1: a <= b and
A2: c <= d ; :: thesis: (LSeg |[a,c]|,|[a,d]|) /\ (LSeg |[a,c]|,|[b,c]|) = {|[a,c]|}
for ax being set holds
( ax in (LSeg |[a,c]|,|[a,d]|) /\ (LSeg |[a,c]|,|[b,c]|) iff ax = |[a,c]| )

proof

let ax be set ; :: thesis: ( ax in (LSeg |[a,c]|,|[a,d]|) /\ (LSeg |[a,c]|,|[b,c]|) iff ax = |[a,c]| )
thus ( ax in (LSeg |[a,c]|,|[a,d]|) /\ (LSeg |[a,c]|,|[b,c]|) implies ax = |[a,c]| ) :: thesis: ( ax = |[a,c]| implies ax in (LSeg |[a,c]|,|[a,d]|) /\ (LSeg |[a,c]|,|[b,c]|) )

proof

assume A3: ax in (LSeg |[a,c]|,|[a,d]|) /\ (LSeg |[a,c]|,|[b,c]|) ; :: thesis: ax = |[a,c]|
then A4: ax in LSeg |[a,c]|,|[a,d]| by XBOOLE_0:def 4;
ax in LSeg |[a,c]|,|[b,c]| by A3, XBOOLE_0:def 4;
then ax in { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 <= b & p2 `1 >= a & p2 `2 = c ) } by A1, Th39;
then A5: ex p2 being Point of (TOP-REAL 2) st
( p2 = ax & p2 `1 <= b & p2 `1 >= a & p2 `2 = c ) ;
ax in { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 = a & p2 `2 <= d & p2 `2 >= c ) } by A2, A4, Th39;
then ex p being Point of (TOP-REAL 2) st
( p = ax & p `1 = a & p `2 <= d & p `2 >= c ) ;
hence ax = |[a,c]| by A5, EUCLID:57; :: thesis: verum

end;

assume A6: ax = |[a,c]| ; :: thesis: ax in (LSeg |[a,c]|,|[a,d]|) /\ (LSeg |[a,c]|,|[b,c]|)
then A7: ax in LSeg |[a,c]|,|[a,d]| by RLTOPSP1:69;
ax in LSeg |[a,c]|,|[b,c]| by A6, RLTOPSP1:69;
hence ax in (LSeg |[a,c]|,|[a,d]|) /\ (LSeg |[a,c]|,|[b,c]|) by A7, XBOOLE_0:def 4; :: thesis: verum

end;

hence (LSeg |[a,c]|,|[a,d]|) /\ (LSeg |[a,c]|,|[b,c]|) = {|[a,c]|} by TARSKI:def 1; :: thesis: verum