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:
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)) ;
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 not 0. (TOP-REAL n) in LSeg w1,w4 by RLTOPSP1:72;
then consider w0 being Point of (TOP-REAL n) such that
w0 in LSeg w1,w4 and
A2: |.w0.| > 0 and
A3: |.w0.| = (dist_min P) . (0. (TOP-REAL n)) by Th48;
set l9 = 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 reconsider q2 = z as Point of (TOP-REAL n) by EUCLID:25;
not z in { q where q is Point of (TOP-REAL n) : |.q.| < a } by A5, XBOOLE_0:def 5;
then |.q2.| >= a ;
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
A6: z = q1 and
A7: |.q1.| >= a ;
q1 in the carrier of (TOP-REAL n) ;
then A8: z in REAL n by A6, EUCLID:25;
for q being Point of (TOP-REAL n) st q = z holds
|.q.| >= a by A6, A7;
then not z in { q where q is Point of (TOP-REAL n) : |.q.| < a } ;
hence z in (REAL n) \ { q where q is Point of (TOP-REAL n) : |.q.| < a } by A8, XBOOLE_0:def 5; :: thesis:

end;

A9: 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 consider r being Real such that
A10: x = ((1 - r) * w1) + (r * ((a / |.w0.|) * w1)) and
A11: 0 <= r and
A12: r <= 1 ;

now

per cases ;

case A13: a > 0 ; :: thesis:

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)) ;
reconsider o = 0. (TOP-REAL n) as Point of (Euclid n) by EUCLID:71;
reconsider w5 = ((1 - 0 ) * w1) + (0 * w4) as Point of (TOP-REAL n) ;
A14: ((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 ;
(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
z in dom (dist o) and
A15: z in P and
A16: x = (dist o) . z by FUNCT_1:def 12;
reconsider z2 = z as Point of (Euclid n) by A15, TOPREAL3:13;
(dist o) . z2 = dist z2,o by WEIERSTR:def 6;
hence x in REAL by A16; :: thesis:

end;

then reconsider F = (dist o) .: P as Subset of REAL ;
reconsider w59 = w5 as Point of (Euclid n) by TOPREAL3:13;
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
x in dom (dist o) and
A17: x in P and
A18: r = (dist o) . x by FUNCT_1:def 12;
reconsider w0 = x as Point of (Euclid n) by A17, TOPREAL3:13;
r = dist w0,o by A18, WEIERSTR:def 6;
hence 0 <= r by METRIC_1:5; :: thesis:

end;

then A19: F is bounded_below by SEQ_4:def 2;
TopStruct(# the carrier of (TOP-REAL n),the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def 8;
then w59 in the carrier of (TopSpaceMetr (Euclid n)) ;
then A20: w59 in dom (dist o) by FUNCT_2:def 1;
( w5 in LSeg w1,w4 & dist w59,o = (dist o) . w59 ) by WEIERSTR:def 6;
then dist w59,o in (dist o) .: P by A20, FUNCT_1:def 12;
then lower_bound F <= dist w59,o by A19, SEQ_4:def 5;
then dist w59,o >= lower_bound ([#] ((dist o) .: P)) by WEIERSTR:def 3;
then dist w59,o >= lower_bound ((dist o) .: P) by WEIERSTR:def 5;
then dist w59,o >= |.w0.| by A3, WEIERSTR:def 8;
then |.(w5 - (0. (TOP-REAL n))).| >= |.w0.| by JGRAPH_1:45;
then A21: |.w5.| >= |.w0.| by RLVECT_1:26;
A22: 1 - r >= 0 by A12, XREAL_1:50;
then A23: (abs ((1 - r) + (r * (a / |.w0.|)))) * |.w1.| = ((1 - r) + (r * (a / |.w0.|))) * |.w1.| by A11, A13, ABSVALUE:def 1
.= ((1 - r) * |.w1.|) + ((r * (a / |.w0.|)) * |.w1.|) ;
ex q1 being Point of (TOP-REAL n) st
( q1 = w1 & |.q1.| >= a ) by A1, A4;
then A24: (1 - r) * |.w1.| >= (1 - r) * a by A22, XREAL_1:66;
(r * (a / |.w0.|)) * |.w0.| = ((r * a) / |.w0.|) * |.w0.| by XCMPLX_1:75
.= r * a by A2, XCMPLX_1:88 ;
then (r * (a / |.w0.|)) * |.w1.| >= r * a by A11, A13, A14, A21, XREAL_1:66;
then (abs ((1 - r) + (r * (a / |.w0.|)))) * |.w1.| >= (r * a) + ((1 - r) * a) by A24, A23, 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, A10; :: thesis:

end;

A25: 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 consider r being Real such that
A26: x = ((1 - r) * w4) + (r * ((a / |.w0.|) * w4)) and
A27: 0 <= r and
A28: r <= 1 ;

now

per cases ;

case A29: a > 0 ; :: thesis:

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)) ;
reconsider o = 0. (TOP-REAL n) as Point of (Euclid n) by EUCLID:71;
reconsider w5 = ((1 - 0 ) * w4) + (0 * w1) as Point of (TOP-REAL n) ;
A30: ((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 ;
(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
z in dom (dist o) and
A31: z in P and
A32: x = (dist o) . z by FUNCT_1:def 12;
reconsider z2 = z as Point of (Euclid n) by A31, TOPREAL3:13;
(dist o) . z2 = dist z2,o by WEIERSTR:def 6;
hence x in REAL by A32; :: thesis:

end;

then reconsider F = (dist o) .: P as Subset of REAL ;
reconsider w59 = w5 as Point of (Euclid n) by TOPREAL3:13;
A33: dist w59,o = (dist o) . w59 by WEIERSTR:def 6;
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
x in dom (dist o) and
A34: x in P and
A35: r = (dist o) . x by FUNCT_1:def 12;
reconsider w0 = x as Point of (Euclid n) by A34, TOPREAL3:13;
r = dist w0,o by A35, WEIERSTR:def 6;
hence 0 <= r by METRIC_1:5; :: thesis:

end;

then A36: F is bounded_below by SEQ_4:def 2;
TopStruct(# the carrier of (TOP-REAL n),the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def 8;
then w59 in the carrier of (TopSpaceMetr (Euclid n)) ;
then A37: w59 in dom (dist o) by FUNCT_2:def 1;
w5 in { (((1 - r1) * w4) + (r1 * w1)) where r1 is Real : ( 0 <= r1 & r1 <= 1 ) } ;
then dist w59,o in (dist o) .: P by A37, A33, FUNCT_1:def 12;
then lower_bound F <= dist w59,o by A36, SEQ_4:def 5;
then dist w59,o >= lower_bound ([#] ((dist o) .: P)) by WEIERSTR:def 3;
then dist w59,o >= lower_bound ((dist o) .: P) by WEIERSTR:def 5;
then dist w59,o >= |.w0.| by A3, WEIERSTR:def 8;
then |.(w5 - (0. (TOP-REAL n))).| >= |.w0.| by JGRAPH_1:45;
then A38: |.w5.| >= |.w0.| by RLVECT_1:26;
A39: 1 - r >= 0 by A28, XREAL_1:50;
then A40: (abs ((1 - r) + (r * (a / |.w0.|)))) * |.w4.| = ((1 - r) + (r * (a / |.w0.|))) * |.w4.| by A27, A29, ABSVALUE:def 1
.= ((1 - r) * |.w4.|) + ((r * (a / |.w0.|)) * |.w4.|) ;
ex q1 being Point of (TOP-REAL n) st
( q1 = w4 & |.q1.| >= a ) by A1, A4;
then A41: (1 - r) * |.w4.| >= (1 - r) * a by A39, XREAL_1:66;
(r * (a / |.w0.|)) * |.w0.| = ((r * a) / |.w0.|) * |.w0.| by XCMPLX_1:75
.= r * a by A2, XCMPLX_1:88 ;
then (r * (a / |.w0.|)) * |.w4.| >= r * a by A27, A29, A30, A38, XREAL_1:66;
then (abs ((1 - r) + (r * (a / |.w0.|)))) * |.w4.| >= (r * a) + ((1 - r) * a) by A41, A40, 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, A26; :: thesis:

end;

A42: LSeg ((a / |.w0.|) * w1),((a / |.w0.|) * w4) c= Q

proof

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)) ;
reconsider o = 0. (TOP-REAL n) as Point of (Euclid n) by EUCLID:71;
let x be set ; :: according to TARSKI:def 3 :: thesis:
A43: abs (a / |.w0.|) = (abs a) / (abs |.w0.|) by COMPLEX1:153
.= (abs a) / |.w0.| by ABSVALUE:def 1 ;
(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
z in dom (dist o) and
A44: z in P and
A45: x = (dist o) . z by FUNCT_1:def 12;
reconsider z2 = z as Point of (Euclid n) by A44, TOPREAL3:13;
(dist o) . z2 = dist z2,o by WEIERSTR:def 6;
hence x in REAL by A45; :: thesis:

end;

then reconsider F = (dist o) .: P as Subset of REAL ;
assume x in LSeg ((a / |.w0.|) * w1),((a / |.w0.|) * w4) ; :: thesis:
then consider r being Real such that
A46: x = ((1 - r) * ((a / |.w0.|) * w1)) + (r * ((a / |.w0.|) * w4)) and
A47: ( 0 <= r & r <= 1 ) ;
reconsider w5 = ((1 - r) * w1) + (r * w4) as Point of (TOP-REAL n) ;
reconsider w59 = w5 as Point of (Euclid n) by TOPREAL3:13;
A48: dist w59,o = (dist o) . w59 by WEIERSTR:def 6;
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
x in dom (dist o) and
A49: x in P and
A50: r = (dist o) . x by FUNCT_1:def 12;
reconsider w0 = x as Point of (Euclid n) by A49, TOPREAL3:13;
r = dist w0,o by A50, WEIERSTR:def 6;
hence 0 <= r by METRIC_1:5; :: thesis:

end;

then A51: F is bounded_below by SEQ_4:def 2;
TopStruct(# the carrier of (TOP-REAL n),the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def 8;
then w59 in the carrier of (TopSpaceMetr (Euclid n)) ;
then A52: w59 in dom (dist o) by FUNCT_2:def 1;
w5 in LSeg w1,w4 by A47;
then dist w59,o in (dist o) .: P by A52, A48, FUNCT_1:def 12;
then lower_bound F <= dist w59,o by A51, SEQ_4:def 5;
then dist w59,o >= lower_bound ([#] ((dist o) .: P)) by WEIERSTR:def 3;
then dist w59,o >= lower_bound ((dist o) .: P) by WEIERSTR:def 5;
then dist w59,o >= |.w0.| by A3, WEIERSTR:def 8;
then |.(w5 - (0. (TOP-REAL n))).| >= |.w0.| by JGRAPH_1:45;
then |.w5.| >= |.w0.| by RLVECT_1:26;
then ( abs a >= 0 & |.w5.| / |.w0.| >= 1 ) by A2, COMPLEX1:132, XREAL_1:183;
then (abs a) * (|.w5.| / |.w0.|) >= (abs a) * 1 by XREAL_1:68;
then (abs a) * (|.w5.| * (|.w0.| " )) >= abs a by XCMPLX_0:def 9;
then ((abs a) * (|.w0.| " )) * |.w5.| >= abs a ;
then A53: ((abs a) / |.w0.|) * |.w5.| >= abs a by XCMPLX_0:def 9;
abs a >= a by ABSVALUE:11;
then ((abs a) / |.w0.|) * |.w5.| >= a by A53, XXREAL_0:2;
then |.((a / |.w0.|) * (((1 - r) * w1) + (r * w4))).| >= a by A43, 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, A46; :: thesis:

end;

( (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;
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 A42, A9, A25; :: thesis: