let n be Element of NAT ; :: thesis:
let a be Real; :: thesis:
let Q be Subset of (TOP-REAL n); :: thesis:
let w1, w4 be Point of (TOP-REAL n); :: thesis:
assume A1: ( Q = (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } & w1 in Q & w4 in Q & ( for r being Real holds
( not w1 = r * w4 & not w4 = r * w1 ) ) ) ; :: thesis:
then A2: not 0. (TOP-REAL n) in LSeg w1,w4 by RLTOPSP1:72;
V: TopStruct(# the carrier of (TOP-REAL n),the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def 8;
the carrier of (TOP-REAL n) = the carrier of (Euclid n) by EUCLID:71;
then reconsider P = LSeg w1,w4 as Subset of (TopSpaceMetr (Euclid n)) by XX, YY, V;
consider w0 being Point of (TOP-REAL n) such that
A3: ( w0 in LSeg w1,w4 & |.w0.| > 0 & |.w0.| = (dist_min P) . (0. (TOP-REAL n)) ) by A2, Th48;
set l' = a / |.w0.|;
set w2 = (a / |.w0.|) * w1;
set w3 = (a / |.w0.|) * w4;
A4: (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } = { q1 where q1 is Point of (TOP-REAL n) : |.q1.| >= a }

proof

thus (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } c= { q1 where q1 is Point of (TOP-REAL n) : |.q1.| >= a } :: according to XBOOLE_0:def 10 :: thesis:

proof

let z be set ; :: according to TARSKI:def 3 :: thesis:
assume A5: z in (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } ; :: thesis:
then A6: ( z in REAL n & not z in { q where q is Point of (TOP-REAL n) : |.q.| < a } ) by XBOOLE_0:def 5;
reconsider q2 = z as Point of (TOP-REAL n) by A5, EUCLID:25;
|.q2.| >= a by A6;
hence z in { q1 where q1 is Point of (TOP-REAL n) : |.q1.| >= a } ; :: thesis:

end;

let z be set ; :: according to TARSKI:def 3 :: thesis:
assume z in { q1 where q1 is Point of (TOP-REAL n) : |.q1.| >= a } ; :: thesis:
then consider q1 being Point of (TOP-REAL n) such that
A7: ( z = q1 & |.q1.| >= a ) ;
A8: q1 in the carrier of (TOP-REAL n) ;
for q being Point of (TOP-REAL n) st q = z holds
|.q.| >= a by A7;
then ( z in REAL n & not z in { q where q is Point of (TOP-REAL n) : |.q.| < a } ) by A7, A8, EUCLID:25;
hence z in (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } by XBOOLE_0:def 5; :: thesis:

end;

A9: LSeg ((a / |.w0.|) * w1),((a / |.w0.|) * w4) c= Q

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in LSeg ((a / |.w0.|) * w1),((a / |.w0.|) * w4) ; :: thesis:
then x in { (((1 - r) * ((a / |.w0.|) * w1)) + (r * ((a / |.w0.|) * w4))) where r is Real : ( 0 <= r & r <= 1 ) } ;
then consider r being Real such that
A10: ( x = ((1 - r) * ((a / |.w0.|) * w1)) + (r * ((a / |.w0.|) * w4)) & 0 <= r & r <= 1 ) ;
reconsider w5 = ((1 - r) * w1) + (r * w4) as Point of (TOP-REAL n) ;
reconsider w5' = w5 as Point of (Euclid n) by TOPREAL3:13;
w5 in { (((1 - r1) * w1) + (r1 * w4)) where r1 is Real : ( 0 <= r1 & r1 <= 1 ) } by A10;
then A11: w5 in LSeg w1,w4 ;
V: TopStruct(# the carrier of (TOP-REAL n),the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def 8;
U: the carrier of (TOP-REAL n) = the carrier of (Euclid n) by EUCLID:71;
then reconsider o = 0. (TOP-REAL n) as Point of (Euclid n) by XX, YY;
reconsider P = LSeg w1,w4 as Subset of (TopSpaceMetr (Euclid n)) by XX, YY, U, V;
(dist o) .: P c= REAL

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in (dist o) .: P ; :: thesis:
then consider z being set such that
A12: ( z in dom (dist o) & z in P & x = (dist o) . z ) by FUNCT_1:def 12;
reconsider z2 = z as Point of (Euclid n) by A12, TOPREAL3:13;
(dist o) . z2 = dist z2,o by WEIERSTR:def 6;
hence x in REAL by A12; :: thesis:

end;

then reconsider F = (dist o) .: P as Subset of REAL ;
V: TopStruct(# the carrier of (TOP-REAL n),the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def 8;
the carrier of (TOP-REAL n) = the carrier of (Euclid n) by EUCLID:71;
then A13: w5' in the carrier of (TopSpaceMetr (Euclid n)) by XX, YY, V;
for r being real number st r in (dist o) .: P holds
0 <= r

proof

let r be real number ; :: thesis:
assume r in (dist o) .: P ; :: thesis:
then consider x being set such that
A14: ( x in dom (dist o) & x in P & r = (dist o) . x ) by FUNCT_1:def 12;
reconsider w0 = x as Point of (Euclid n) by A14, TOPREAL3:13;
r = dist w0,o by A14, WEIERSTR:def 6;
hence 0 <= r by METRIC_1:5; :: thesis:

end;

then A15: F is bounded_below by SEQ_4:def 2;
( w5' in dom (dist o) & dist w5',o = (dist o) . w5' ) by A13, FUNCT_2:def 1, WEIERSTR:def 6;
then dist w5',o in (dist o) .: P by A11, FUNCT_1:def 12;
then lower_bound F <= dist w5',o by A15, SEQ_4:def 5;
then dist w5',o >= lower_bound ([#] ((dist o) .: P)) by WEIERSTR:def 3;
then dist w5',o >= lower_bound ((dist o) .: P) by WEIERSTR:def 5;
then dist w5',o >= |.w0.| by A3, WEIERSTR:def 8;
then |.(w5 - (0. (TOP-REAL n))).| >= |.w0.| by JGRAPH_1:45;
then A16: |.w5.| >= |.w0.| by RLVECT_1:26, RLVECT_1:27;
A17: abs a >= 0 by COMPLEX1:132;
A18: abs (a / |.w0.|) = (abs a) / (abs |.w0.|) by COMPLEX1:153
.= (abs a) / |.w0.| by ABSVALUE:def 1 ;
|.w5.| / |.w0.| >= 1 by A3, A16, XREAL_1:183;
then (abs a) * (|.w5.| / |.w0.|) >= (abs a) * 1 by A17, XREAL_1:68;
then (abs a) * (|.w5.| * (|.w0.| " )) >= abs a by XCMPLX_0:def 9;
then ((abs a) * (|.w0.| " )) * |.w5.| >= abs a ;
then A19: ((abs a) / |.w0.|) * |.w5.| >= abs a by XCMPLX_0:def 9;
abs a >= a by ABSVALUE:11;
then ((abs a) / |.w0.|) * |.w5.| >= a by A19, XXREAL_0:2;
then |.((a / |.w0.|) * (((1 - r) * w1) + (r * w4))).| >= a by A18, TOPRNS_1:8;
then |.(((a / |.w0.|) * ((1 - r) * w1)) + ((a / |.w0.|) * (r * w4))).| >= a by EUCLID:36;
then |.(((a / |.w0.|) * ((1 - r) * w1)) + (((a / |.w0.|) * r) * w4)).| >= a by EUCLID:34;
then |.((((a / |.w0.|) * (1 - r)) * w1) + (((a / |.w0.|) * r) * w4)).| >= a by EUCLID:34;
then |.((((1 - r) * (a / |.w0.|)) * w1) + (r * ((a / |.w0.|) * w4))).| >= a by EUCLID:34;
then |.(((1 - r) * ((a / |.w0.|) * w1)) + (r * ((a / |.w0.|) * w4))).| >= a by EUCLID:34;
hence x in Q by A1, A4, A10; :: thesis:

end;

A20: ( (a / |.w0.|) * w1 in LSeg ((a / |.w0.|) * w1),((a / |.w0.|) * w4) & (a / |.w0.|) * w4 in LSeg ((a / |.w0.|) * w1),((a / |.w0.|) * w4) ) by RLTOPSP1:69;
A21: LSeg w1,((a / |.w0.|) * w1) c= Q

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in LSeg w1,((a / |.w0.|) * w1) ; :: thesis:
then x in { (((1 - r) * w1) + (r * ((a / |.w0.|) * w1))) where r is Real : ( 0 <= r & r <= 1 ) } ;
then consider r being Real such that
A22: ( x = ((1 - r) * w1) + (r * ((a / |.w0.|) * w1)) & 0 <= r & r <= 1 ) ;

now

per cases ;

case a > 0 ; :: thesis:

then a / |.w0.| > 0 by A3, XREAL_1:141;
then A23: r * (a / |.w0.|) >= 0 by A22;
reconsider w5 = ((1 - 0 ) * w1) + (0 * w4) as Point of (TOP-REAL n) ;
A24: ((1 - 0 ) * w1) + (0 * w4) = ((1 - 0 ) * w1) + (0. (TOP-REAL n)) by EUCLID:33
.= (1 - 0 ) * w1 by EUCLID:31
.= w1 by EUCLID:33 ;
reconsider w5' = w5 as Point of (Euclid n) by TOPREAL3:13;
w5 in { (((1 - r1) * w1) + (r1 * w4)) where r1 is Real : ( 0 <= r1 & r1 <= 1 ) } ;
then A25: w5 in LSeg w1,w4 ;
U: the carrier of (TOP-REAL n) = the carrier of (Euclid n) by EUCLID:71;
then reconsider o = 0. (TOP-REAL n) as Point of (Euclid n) by XX, YY;
V: TopStruct(# the carrier of (TOP-REAL n),the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def 8;
then reconsider P = LSeg w1,w4 as Subset of (TopSpaceMetr (Euclid n)) by XX, YY, U;
(dist o) .: P c= REAL

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in (dist o) .: P ; :: thesis:
then consider z being set such that
A26: ( z in dom (dist o) & z in P & x = (dist o) . z ) by FUNCT_1:def 12;
reconsider z2 = z as Point of (Euclid n) by A26, TOPREAL3:13;
(dist o) . z2 = dist z2,o by WEIERSTR:def 6;
hence x in REAL by A26; :: thesis:

end;

then reconsider F = (dist o) .: P as Subset of REAL ;
V: TopStruct(# the carrier of (TOP-REAL n),the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def 8;
the carrier of (TOP-REAL n) = the carrier of (Euclid n) by EUCLID:71;
then A27: w5' in the carrier of (TopSpaceMetr (Euclid n)) by XX, YY, V;
for r being real number st r in (dist o) .: P holds
0 <= r

proof

let r be real number ; :: thesis:
assume r in (dist o) .: P ; :: thesis:
then consider x being set such that
A28: ( x in dom (dist o) & x in P & r = (dist o) . x ) by FUNCT_1:def 12;
reconsider w0 = x as Point of (Euclid n) by A28, TOPREAL3:13;
r = dist w0,o by A28, WEIERSTR:def 6;
hence 0 <= r by METRIC_1:5; :: thesis:

end;

then A29: F is bounded_below by SEQ_4:def 2;
( w5' in dom (dist o) & dist w5',o = (dist o) . w5' ) by A27, FUNCT_2:def 1, WEIERSTR:def 6;
then dist w5',o in (dist o) .: P by A25, FUNCT_1:def 12;
then lower_bound F <= dist w5',o by A29, SEQ_4:def 5;
then dist w5',o >= lower_bound ([#] ((dist o) .: P)) by WEIERSTR:def 3;
then dist w5',o >= lower_bound ((dist o) .: P) by WEIERSTR:def 5;
then dist w5',o >= |.w0.| by A3, WEIERSTR:def 8;
then |.(w5 - (0. (TOP-REAL n))).| >= |.w0.| by JGRAPH_1:45;
then A30: |.w5.| >= |.w0.| by RLVECT_1:26, RLVECT_1:27;
(r * (a / |.w0.|)) * |.w0.| = ((r * a) / |.w0.|) * |.w0.| by XCMPLX_1:75
.= r * a by A3, XCMPLX_1:88 ;
then A31: (r * (a / |.w0.|)) * |.w1.| >= r * a by A23, A24, A30, XREAL_1:66;
consider q1 being Point of (TOP-REAL n) such that
A32: ( q1 = w1 & |.q1.| >= a ) by A1, A4;
A33: 1 - r >= 0 by A22, XREAL_1:50;
then A34: (1 - r) * |.w1.| >= (1 - r) * a by A32, XREAL_1:66;
(1 - r) + (r * (a / |.w0.|)) >= 0 by A23, A33;
then (abs ((1 - r) + (r * (a / |.w0.|)))) * |.w1.| = ((1 - r) + (r * (a / |.w0.|))) * |.w1.| by ABSVALUE:def 1
.= ((1 - r) * |.w1.|) + ((r * (a / |.w0.|)) * |.w1.|) ;
then (abs ((1 - r) + (r * (a / |.w0.|)))) * |.w1.| >= (r * a) + ((1 - r) * a) by A31, A34, XREAL_1:9;
then |.(((1 - r) + (r * (a / |.w0.|))) * w1).| >= a by TOPRNS_1:8;
then |.(((1 - r) * w1) + ((r * (a / |.w0.|)) * w1)).| >= a by EUCLID:37;
hence |.(((1 - r) * w1) + (r * ((a / |.w0.|) * w1))).| >= a by EUCLID:34; :: thesis:

end;

case a <= 0 ; :: thesis:

hence |.(((1 - r) * w1) + (r * ((a / |.w0.|) * w1))).| >= a ; :: thesis:

end;

hence x in Q by A1, A4, A22; :: thesis:

end;

LSeg w4,((a / |.w0.|) * w4) c= Q

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in LSeg w4,((a / |.w0.|) * w4) ; :: thesis:
then x in { (((1 - r) * w4) + (r * ((a / |.w0.|) * w4))) where r is Real : ( 0 <= r & r <= 1 ) } ;
then consider r being Real such that
A35: ( x = ((1 - r) * w4) + (r * ((a / |.w0.|) * w4)) & 0 <= r & r <= 1 ) ;

now

per cases ;

case a > 0 ; :: thesis:

then a / |.w0.| > 0 by A3, XREAL_1:141;
then A36: r * (a / |.w0.|) >= 0 by A35;
reconsider w5 = ((1 - 0 ) * w4) + (0 * w1) as Point of (TOP-REAL n) ;
A37: ((1 - 0 ) * w4) + (0 * w1) = ((1 - 0 ) * w4) + (0. (TOP-REAL n)) by EUCLID:33
.= (1 - 0 ) * w4 by EUCLID:31
.= w4 by EUCLID:33 ;
reconsider w5' = w5 as Point of (Euclid n) by TOPREAL3:13;
w5 in { (((1 - r1) * w4) + (r1 * w1)) where r1 is Real : ( 0 <= r1 & r1 <= 1 ) } ;
then A38: w5 in LSeg w1,w4 by RLTOPSP1:def 2;
U: the carrier of (TOP-REAL n) = the carrier of (Euclid n) by EUCLID:71;
then reconsider o = 0. (TOP-REAL n) as Point of (Euclid n) by XX, YY;
V: TopStruct(# the carrier of (TOP-REAL n),the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def 8;
then reconsider P = LSeg w4,w1 as Subset of (TopSpaceMetr (Euclid n)) by XX, YY, U;
(dist o) .: P c= REAL

proof

let x be set ; :: according to TARSKI:def 3 :: thesis:
assume x in (dist o) .: P ; :: thesis:
then consider z being set such that
A39: ( z in dom (dist o) & z in P & x = (dist o) . z ) by FUNCT_1:def 12;
reconsider z2 = z as Point of (Euclid n) by A39, TOPREAL3:13;
(dist o) . z2 = dist z2,o by WEIERSTR:def 6;
hence x in REAL by A39; :: thesis:

end;

then reconsider F = (dist o) .: P as Subset of REAL ;
V: TopStruct(# the carrier of (TOP-REAL n),the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def 8;
the carrier of (TOP-REAL n) = the carrier of (Euclid n) by EUCLID:71;
then A40: w5' in the carrier of (TopSpaceMetr (Euclid n)) by XX, YY, V;
for r being real number st r in (dist o) .: P holds
0 <= r

proof

let r be real number ; :: thesis:
assume r in (dist o) .: P ; :: thesis:
then consider x being set such that
A41: ( x in dom (dist o) & x in P & r = (dist o) . x ) by FUNCT_1:def 12;
reconsider w0 = x as Point of (Euclid n) by A41, TOPREAL3:13;
r = dist w0,o by A41, WEIERSTR:def 6;
hence 0 <= r by METRIC_1:5; :: thesis:

end;

then A42: F is bounded_below by SEQ_4:def 2;
( w5' in dom (dist o) & dist w5',o = (dist o) . w5' ) by A40, FUNCT_2:def 1, WEIERSTR:def 6;
then dist w5',o in (dist o) .: P by A38, FUNCT_1:def 12;
then lower_bound F <= dist w5',o by A42, SEQ_4:def 5;
then dist w5',o >= lower_bound ([#] ((dist o) .: P)) by WEIERSTR:def 3;
then dist w5',o >= lower_bound ((dist o) .: P) by WEIERSTR:def 5;
then dist w5',o >= |.w0.| by A3, WEIERSTR:def 8;
then |.(w5 - (0. (TOP-REAL n))).| >= |.w0.| by JGRAPH_1:45;
then A43: |.w5.| >= |.w0.| by RLVECT_1:26, RLVECT_1:27;
(r * (a / |.w0.|)) * |.w0.| = ((r * a) / |.w0.|) * |.w0.| by XCMPLX_1:75
.= r * a by A3, XCMPLX_1:88 ;
then A44: (r * (a / |.w0.|)) * |.w4.| >= r * a by A36, A37, A43, XREAL_1:66;
consider q1 being Point of (TOP-REAL n) such that
A45: ( q1 = w4 & |.q1.| >= a ) by A1, A4;
A46: 1 - r >= 0 by A35, XREAL_1:50;
then A47: (1 - r) * |.w4.| >= (1 - r) * a by A45, XREAL_1:66;
(1 - r) + (r * (a / |.w0.|)) >= 0 by A36, A46;
then (abs ((1 - r) + (r * (a / |.w0.|)))) * |.w4.| = ((1 - r) + (r * (a / |.w0.|))) * |.w4.| by ABSVALUE:def 1
.= ((1 - r) * |.w4.|) + ((r * (a / |.w0.|)) * |.w4.|) ;
then (abs ((1 - r) + (r * (a / |.w0.|)))) * |.w4.| >= (r * a) + ((1 - r) * a) by A44, A47, XREAL_1:9;
then |.(((1 - r) + (r * (a / |.w0.|))) * w4).| >= a by TOPRNS_1:8;
then |.(((1 - r) * w4) + ((r * (a / |.w0.|)) * w4)).| >= a by EUCLID:37;
hence |.(((1 - r) * w4) + (r * ((a / |.w0.|) * w4))).| >= a by EUCLID:34; :: thesis:

end;

case a <= 0 ; :: thesis:

hence |.(((1 - r) * w4) + (r * ((a / |.w0.|) * w4))).| >= a ; :: thesis:

end;

hence x in Q by A1, A4, A35; :: thesis:

end;

hence ex w2, w3 being Point of (TOP-REAL n) st
( w2 in Q & w3 in Q & LSeg w1,w2 c= Q & LSeg w2,w3 c= Q & LSeg w3,w4 c= Q ) by A9, A20, A21; :: thesis: