:: Sequences of Metric Spaces and an Abstract Intermediate Value Theorem
:: by Yatsuka Nakamura and Andrzej Trybulec
::
:: Received September 11, 2002
:: Copyright (c) 2002-2021 Association of Mizar Users

theorem Th1: :: TOPMETR3:1
for X being non empty MetrSpace
for S being sequence of X
for F being Subset of () st S is convergent & ( for n being Nat holds S . n in F ) & F is closed holds
lim S in F
proof end;

theorem Th2: :: TOPMETR3:2
for X, Y being non empty MetrSpace
for f being Function of (),()
for S being sequence of X holds f * S is sequence of Y
proof end;

theorem Th3: :: TOPMETR3:3
for X, Y being non empty MetrSpace
for f being Function of (),()
for S being sequence of X
for T being sequence of Y st S is convergent & T = f * S & f is continuous holds
T is convergent
proof end;

theorem Th4: :: TOPMETR3:4
for s being Real_Sequence
for S being sequence of RealSpace st s = S holds
( ( s is convergent implies S is convergent ) & ( S is convergent implies s is convergent ) & ( s is convergent implies lim s = lim S ) )
proof end;

theorem Th5: :: TOPMETR3:5
for a, b being Real
for s being Real_Sequence st rng s c= [.a,b.] holds
s is sequence of ()
proof end;

theorem Th6: :: TOPMETR3:6
for a, b being Real
for S being sequence of () st a <= b holds
S is sequence of RealSpace
proof end;

theorem Th7: :: TOPMETR3:7
for a, b being Real
for S1 being sequence of ()
for S being sequence of RealSpace st S = S1 & a <= b holds
( ( S is convergent implies S1 is convergent ) & ( S1 is convergent implies S is convergent ) & ( S is convergent implies lim S = lim S1 ) )
proof end;

theorem Th8: :: TOPMETR3:8
for a, b being Real
for s being Real_Sequence
for S being sequence of () st S = s & a <= b & s is convergent holds
( S is convergent & lim s = lim S )
proof end;

theorem :: TOPMETR3:9
for a, b being Real
for s being Real_Sequence
for S being sequence of () st S = s & a <= b & s is non-decreasing holds
S is convergent
proof end;

theorem :: TOPMETR3:10
for a, b being Real
for s being Real_Sequence
for S being sequence of () st S = s & a <= b & s is non-increasing holds
S is convergent
proof end;

theorem Th11: :: TOPMETR3:11
for R being non empty Subset of REAL st R is bounded_above holds
ex s being Real_Sequence st
( s is non-decreasing & s is convergent & rng s c= R & lim s = upper_bound R )
proof end;

theorem Th12: :: TOPMETR3:12
for R being non empty Subset of REAL st R is bounded_below holds
ex s being Real_Sequence st
( s is non-increasing & s is convergent & rng s c= R & lim s = lower_bound R )
proof end;

theorem Th13: :: TOPMETR3:13
for X being non empty MetrSpace
for f being Function of I[01],()
for F1, F2 being Subset of ()
for r1, r2 being Real st 0 <= r1 & r2 <= 1 & r1 <= r2 & f . r1 in F1 & f . r2 in F2 & F1 is closed & F2 is closed & f is continuous & F1 \/ F2 = the carrier of X holds
ex r being Real st
( r1 <= r & r <= r2 & f . r in F1 /\ F2 )
proof end;

theorem Th14: :: TOPMETR3:14
for n being Nat
for p1, p2 being Point of ()
for P, P1 being non empty Subset of () st P is_an_arc_of p1,p2 & P1 is_an_arc_of p2,p1 & P1 c= P holds
P1 = P
proof end;

theorem :: TOPMETR3:15
for P, P1 being non empty compact Subset of () st P is being_simple_closed_curve & P1 is_an_arc_of W-min P, E-max P & P1 c= P & not P1 = Upper_Arc P holds
P1 = Lower_Arc P
proof end;