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 A1:
( a <= b & 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
ax in (LSeg |[a,c]|,|[a,d]|) /\ (LSeg |[a,c]|,|[b,c]|)
;
:: thesis: ax = |[a,c]|
then A2:
(
ax in LSeg |[a,c]|,
|[a,d]| &
ax in LSeg |[a,c]|,
|[b,c]| )
by 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 A3:
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 A1, A2, 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 A3, EUCLID:57;
:: thesis: verum
end;
assume
ax = |[a,c]|
;
:: thesis: ax in (LSeg |[a,c]|,|[a,d]|) /\ (LSeg |[a,c]|,|[b,c]|)
then
(
ax in LSeg |[a,c]|,
|[a,d]| &
ax in LSeg |[a,c]|,
|[b,c]| )
by RLTOPSP1:69;
hence
ax in (LSeg |[a,c]|,|[a,d]|) /\ (LSeg |[a,c]|,|[b,c]|)
by 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