begin
Lm1:
( 0 in [.0,1.] & 1 in [.0,1.] )
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem Th4:
theorem Th5:
theorem
theorem Th7:
theorem Th8:
begin
:: deftheorem defines convex JORDAN1:def 1 :
for V being RealLinearSpace
for P being Subset of V holds
( P is convex iff for w1, w2 being Element of V st w1 in P & w2 in P holds
LSeg (w1,w2) c= P );
theorem Th9:
Lm2:
the carrier of (TOP-REAL 2) = REAL 2
by EUCLID:25;
Lm3:
for s1 being Real holds { |[sb,tb]| where sb, tb is Real : sb < s1 } is Subset of (REAL 2)
Lm4:
for t1 being Real holds { |[sb,tb]| where sb, tb is Real : tb < t1 } is Subset of (REAL 2)
Lm5:
for s2 being Real holds { |[sb,tb]| where sb, tb is Real : s2 < sb } is Subset of (REAL 2)
Lm6:
for t2 being Real holds { |[sb,tb]| where sb, tb is Real : t2 < tb } is Subset of (REAL 2)
Lm7:
for s1, s2, t1, t2 being Real holds { |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } is Subset of (REAL 2)
Lm8:
for s1, s2, t1, t2 being Real holds { |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } is Subset of (REAL 2)
theorem
canceled;
theorem
canceled;
theorem Th12:
for
s1,
s2,
t1,
t2 being
Real holds
{ |[s,t]| where s, t is Real : ( s1 < s & s < s2 & t1 < t & t < t2 ) } = (( { |[s3,t3]| where s3, t3 is Real : s1 < s3 } /\ { |[s4,t4]| where s4, t4 is Real : s4 < s2 } ) /\ { |[s5,t5]| where s5, t5 is Real : t1 < t5 } ) /\ { |[s6,t6]| where s6, t6 is Real : t6 < t2 }
theorem Th13:
for
s1,
s2,
t1,
t2 being
Real holds
{ |[s,t]| where s, t is Real : ( not s1 <= s or not s <= s2 or not t1 <= t or not t <= t2 ) } = (( { |[s3,t3]| where s3, t3 is Real : s3 < s1 } \/ { |[s4,t4]| where s4, t4 is Real : t4 < t1 } ) \/ { |[s5,t5]| where s5, t5 is Real : s2 < s5 } ) \/ { |[s6,t6]| where s6, t6 is Real : t2 < t6 }
theorem Th14:
theorem
theorem Th16:
theorem
theorem Th18:
theorem
theorem Th20:
theorem
theorem Th22:
theorem
theorem Th24:
Lm9:
TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2)
by EUCLID:def 8;
theorem Th25:
theorem Th26:
theorem Th27:
theorem Th28:
theorem Th29:
theorem Th30:
theorem Th31:
for
s1,
t1,
s2,
t2 being
Real for
P,
Q being
Subset of
(TOP-REAL 2) st
P = { |[sa,ta]| where sa, ta is Real : ( s1 < sa & sa < s2 & t1 < ta & ta < t2 ) } &
Q = { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) } holds
P misses Q
theorem Th32:
theorem Th33:
for
s1,
s2,
t1,
t2 being
Real holds
{ qc where qc is Point of (TOP-REAL 2) : ( not s1 <= qc `1 or not qc `1 <= s2 or not t1 <= qc `2 or not qc `2 <= t2 ) } = { |[sb,tb]| where sb, tb is Real : ( not s1 <= sb or not sb <= s2 or not t1 <= tb or not tb <= t2 ) }
theorem Th34:
theorem Th35:
theorem Th36:
theorem Th37:
theorem Th38:
theorem Th39:
theorem Th40:
theorem Th41:
Lm10:
for s1, t1, s2, t2 being Real
for P, P1 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P1 = { pa where pa is Point of (TOP-REAL 2) : ( s1 < pa `1 & pa `1 < s2 & t1 < pa `2 & pa `2 < t2 ) } holds
Cl P1 = P \/ P1
Lm11:
for s1, t1, s2, t2 being Real
for P, P2 being Subset of (TOP-REAL 2) st s1 < s2 & t1 < t2 & P = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = s1 & p `2 <= t2 & p `2 >= t1 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t2 ) or ( p `1 <= s2 & p `1 >= s1 & p `2 = t1 ) or ( p `1 = s2 & p `2 <= t2 & p `2 >= t1 ) ) } & P2 = { pb where pb is Point of (TOP-REAL 2) : ( not s1 <= pb `1 or not pb `1 <= s2 or not t1 <= pb `2 or not pb `2 <= t2 ) } holds
Cl P2 = P \/ P2
theorem Th42:
theorem Th43:
theorem
theorem Th45:
theorem
begin
:: deftheorem Def2 defines Jordan JORDAN1:def 2 :
for S being Subset of (TOP-REAL 2) holds
( S is Jordan iff ( S ` <> {} & ex A1, A2 being Subset of (TOP-REAL 2) st
( S ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & ( for C1, C2 being Subset of ((TOP-REAL 2) | (S `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) ) ) ) );
theorem
theorem
theorem
theorem
theorem Th51:
theorem
Lm12:
for x0, y0 being Real
for P being Subset of (TOP-REAL 2) st P = { |[x,y0]| where x is Real : x <= x0 } holds
P is convex
Lm13:
for x0, y0 being Real
for P being Subset of (TOP-REAL 2) st P = { |[x0,y]| where y is Real : y <= y0 } holds
P is convex
Lm14:
for x0, y0 being Real
for P being Subset of (TOP-REAL 2) st P = { |[x,y0]| where x is Real : x >= x0 } holds
P is convex
Lm15:
for x0, y0 being Real
for P being Subset of (TOP-REAL 2) st P = { |[x0,y]| where y is Real : y >= y0 } holds
P is convex