let a, b, c, d be real number ; :: thesis:
set K = rectangle a,b,c,d;
assume that
A1: a < b and
A2: c < d ; :: thesis:
defpred S₁[ set , set ] means for r being Real st $1 = r holds
( ( r in [.0 ,(1 / 2).] implies $2 = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) ) & ( r in [.(1 / 2),1.] implies $2 = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) ) );
A3: [.0 ,1.] = [.0 ,(1 / 2).] \/ [.(1 / 2),1.] by XXREAL_1:165;
A4: for x being set st x in [.0 ,1.] holds
ex y being set st S₁[x,y]

let x be set ; :: thesis:
assume A5: x in [.0 ,1.] ; :: thesis:

per cases by A3, A5, XBOOLE_0:def 3;

case A6: x in [.0 ,(1 / 2).] ; :: thesis:

then reconsider r = x as Real ;
A7: r <= 1 / 2 by A6, XXREAL_1:1;
set y0 = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|);
( r in [.(1 / 2),1.] implies ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) )

proof

assume r in [.(1 / 2),1.] ; :: thesis:
then 1 / 2 <= r by XXREAL_1:1;
then A8: r = 1 / 2 by A7, XXREAL_0:1;
then A9: ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = (0 * |[b,d]|) + |[b,c]| by EUCLID:33
.= (0. (TOP-REAL 2)) + |[b,c]| by EUCLID:33
.= |[b,c]| by EUCLID:31 ;
((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = (1 * |[b,c]|) + (0. (TOP-REAL 2)) by A8, EUCLID:33
.= |[b,c]| + (0. (TOP-REAL 2)) by EUCLID:33
.= |[b,c]| by EUCLID:31 ;
hence ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) by A9; :: thesis:

end;

then for r2 being Real st x = r2 holds
( ( r2 in [.0 ,(1 / 2).] implies ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = ((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|) ) & ( r2 in [.(1 / 2),1.] implies ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = ((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|) ) ) ;
hence ex y being set st S₁[x,y] ; :: thesis:

end;

case A10: x in [.(1 / 2),1.] ; :: thesis:

then reconsider r = x as Real ;
A11: 1 / 2 <= r by A10, XXREAL_1:1;
set y0 = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|);
( r in [.0 ,(1 / 2).] implies ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) )

proof

assume r in [.0 ,(1 / 2).] ; :: thesis:
then r <= 1 / 2 by XXREAL_1:1;
then A12: r = 1 / 2 by A11, XXREAL_0:1;
then A13: ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = |[b,c]| + (0 * |[a,c]|) by EUCLID:33
.= |[b,c]| + (0. (TOP-REAL 2)) by EUCLID:33
.= |[b,c]| by EUCLID:31 ;
((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) = (0. (TOP-REAL 2)) + (1 * |[b,c]|) by A12, EUCLID:33
.= (0. (TOP-REAL 2)) + |[b,c]| by EUCLID:33
.= |[b,c]| by EUCLID:31 ;
hence ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) by A13; :: thesis:

end;

then for r2 being Real st x = r2 holds
( ( r2 in [.0 ,(1 / 2).] implies ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = ((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|) ) & ( r2 in [.(1 / 2),1.] implies ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) = ((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|) ) ) ;
hence ex y being set st S₁[x,y] ; :: thesis:

end;

hence ex y being set st S₁[x,y] ; :: thesis:

end;

ex f2 being Function st
( dom f2 = [.0 ,1.] & ( for x being set st x in [.0 ,1.] holds
S₁[x,f2 . x] ) ) from CLASSES1:sch 1(A4);
then consider f2 being Function such that
A14: dom f2 = [.0 ,1.] and
A15: for x being set st x in [.0 ,1.] holds
S₁[x,f2 . x] ;
rng f2 c= the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle a,b,c,d)))

proof

let y be set ; :: according to TARSKI:def 3 :: thesis:
assume y in rng f2 ; :: thesis:
then consider x being set such that
A16: x in dom f2 and
A17: y = f2 . x by FUNCT_1:def 5;

now

per cases by A3, A14, A16, XBOOLE_0:def 3;

case A18: x in [.0 ,(1 / 2).] ; :: thesis:

then reconsider r = x as Real ;
A19: 0 <= r by A18, XXREAL_1:1;
r <= 1 / 2 by A18, XXREAL_1:1;
then A20: r * 2 <= (1 / 2) * 2 by XREAL_1:66;
f2 . x = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) by A14, A15, A16, A18;
then A21: y in LSeg |[b,d]|,|[b,c]| by A17, A19, A20;
Lower_Arc (rectangle a,b,c,d) = (LSeg |[a,c]|,|[b,c]|) \/ (LSeg |[b,c]|,|[b,d]|) by A1, A2, Th62;
then y in Lower_Arc (rectangle a,b,c,d) by A21, XBOOLE_0:def 3;
hence y in the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle a,b,c,d))) by PRE_TOPC:29; :: thesis:

end;

case A22: x in [.(1 / 2),1.] ; :: thesis:

then reconsider r = x as Real ;
A23: 1 / 2 <= r by A22, XXREAL_1:1;
A24: r <= 1 by A22, XXREAL_1:1;
r * 2 >= (1 / 2) * 2 by A23, XREAL_1:66;
then A25: (2 * r) - 1 >= 0 by XREAL_1:50;
2 * 1 >= 2 * r by A24, XREAL_1:66;
then A26: (1 + 1) - 1 >= (2 * r) - 1 by XREAL_1:11;
f2 . x = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) by A14, A15, A16, A22;
then A27: y in LSeg |[b,c]|,|[a,c]| by A17, A25, A26;
Lower_Arc (rectangle a,b,c,d) = (LSeg |[a,c]|,|[b,c]|) \/ (LSeg |[b,c]|,|[b,d]|) by A1, A2, Th62;
then y in Lower_Arc (rectangle a,b,c,d) by A27, XBOOLE_0:def 3;
hence y in the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle a,b,c,d))) by PRE_TOPC:29; :: thesis:

end;

hence y in the carrier of ((TOP-REAL 2) | (Lower_Arc (rectangle a,b,c,d))) ; :: thesis:

end;

then reconsider f3 = f2 as Function of I[01] ,((TOP-REAL 2) | (Lower_Arc (rectangle a,b,c,d))) by A14, BORSUK_1:83, FUNCT_2:4;
A28: 0 in [.0 ,1.] by XXREAL_1:1;
0 in [.0 ,(1 / 2).] by XXREAL_1:1;
then A29: f3 . 0 = ((1 - (2 * 0 )) * |[b,d]|) + ((2 * 0 ) * |[b,c]|) by A15, A28
.= (1 * |[b,d]|) + (0. (TOP-REAL 2)) by EUCLID:33
.= |[b,d]| + (0. (TOP-REAL 2)) by EUCLID:33
.= |[b,d]| by EUCLID:31
.= E-max (rectangle a,b,c,d) by A1, A2, Th56 ;
A30: 1 in [.0 ,1.] by XXREAL_1:1;
1 in [.(1 / 2),1.] by XXREAL_1:1;
then A31: f3 . 1 = ((1 - ((2 * 1) - 1)) * |[b,c]|) + (((2 * 1) - 1) * |[a,c]|) by A15, A30
.= (0 * |[b,c]|) + |[a,c]| by EUCLID:33
.= (0. (TOP-REAL 2)) + |[a,c]| by EUCLID:33
.= |[a,c]| by EUCLID:31
.= W-min (rectangle a,b,c,d) by A1, A2, Th56 ;
A32: for r being Real st r in [.0 ,(1 / 2).] holds
f3 . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|)

proof

let r be Real; :: thesis:
assume A33: r in [.0 ,(1 / 2).] ; :: thesis:
then A34: 0 <= r by XXREAL_1:1;
r <= 1 / 2 by A33, XXREAL_1:1;
then r <= 1 by XXREAL_0:2;
then r in [.0 ,1.] by A34, XXREAL_1:1;
hence f3 . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) by A15, A33; :: thesis:

end;

A35: for r being Real st r in [.(1 / 2),1.] holds
f3 . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|)

proof

let r be Real; :: thesis:
assume A36: r in [.(1 / 2),1.] ; :: thesis:
then A37: 1 / 2 <= r by XXREAL_1:1;
r <= 1 by A36, XXREAL_1:1;
then r in [.0 ,1.] by A37, XXREAL_1:1;
hence f3 . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) by A15, A36; :: thesis:

end;

A38: for p being Point of (TOP-REAL 2) st p in LSeg |[b,d]|,|[b,c]| holds
( 0 <= (((p `2 ) - d) / (c - d)) / 2 & (((p `2 ) - d) / (c - d)) / 2 <= 1 & f3 . ((((p `2 ) - d) / (c - d)) / 2) = p )

proof

let p be Point of (TOP-REAL 2); :: thesis:
assume A39: p in LSeg |[b,d]|,|[b,c]| ; :: thesis:
A40: |[b,d]| `2 = d by EUCLID:56;
A41: |[b,c]| `2 = c by EUCLID:56;
then A42: c <= p `2 by A2, A39, A40, TOPREAL1:10;
A43: p `2 <= d by A2, A39, A40, A41, TOPREAL1:10;
d - c > 0 by A2, XREAL_1:52;
then A44: - (d - c) < - 0 by XREAL_1:26;
d - (p `2 ) >= 0 by A43, XREAL_1:50;
then A45: - (d - (p `2 )) <= - 0 ;
(p `2 ) - d >= c - d by A42, XREAL_1:11;
then ((p `2 ) - d) / (c - d) <= (c - d) / (c - d) by A44, XREAL_1:75;
then ((p `2 ) - d) / (c - d) <= 1 by A44, XCMPLX_1:60;
then A46: (((p `2 ) - d) / (c - d)) / 2 <= 1 / 2 by XREAL_1:74;
set r = (((p `2 ) - d) / (c - d)) / 2;
(((p `2 ) - d) / (c - d)) / 2 in [.0 ,(1 / 2).] by A44, A45, A46, XXREAL_1:1;
then f3 . ((((p `2 ) - d) / (c - d)) / 2) = ((1 - (2 * ((((p `2 ) - d) / (c - d)) / 2))) * |[b,d]|) + ((2 * ((((p `2 ) - d) / (c - d)) / 2)) * |[b,c]|) by A32
.= |[((1 - (2 * ((((p `2 ) - d) / (c - d)) / 2))) * b),((1 - (2 * ((((p `2 ) - d) / (c - d)) / 2))) * d)]| + ((2 * ((((p `2 ) - d) / (c - d)) / 2)) * |[b,c]|) by EUCLID:62
.= |[((1 - (2 * ((((p `2 ) - d) / (c - d)) / 2))) * b),((1 - (2 * ((((p `2 ) - d) / (c - d)) / 2))) * d)]| + |[((2 * ((((p `2 ) - d) / (c - d)) / 2)) * b),((2 * ((((p `2 ) - d) / (c - d)) / 2)) * c)]| by EUCLID:62
.= |[(((1 * b) - ((2 * ((((p `2 ) - d) / (c - d)) / 2)) * b)) + ((2 * ((((p `2 ) - d) / (c - d)) / 2)) * b)),(((1 - (2 * ((((p `2 ) - d) / (c - d)) / 2))) * d) + ((2 * ((((p `2 ) - d) / (c - d)) / 2)) * c))]| by EUCLID:60
.= |[b,((1 * d) + ((((p `2 ) - d) / (c - d)) * (c - d)))]|
.= |[b,((1 * d) + ((p `2 ) - d))]| by A44, XCMPLX_1:88
.= |[(p `1 ),(p `2 )]| by A39, TOPREAL3:17
.= p by EUCLID:57 ;
hence ( 0 <= (((p `2 ) - d) / (c - d)) / 2 & (((p `2 ) - d) / (c - d)) / 2 <= 1 & f3 . ((((p `2 ) - d) / (c - d)) / 2) = p ) by A44, A45, A46, XXREAL_0:2; :: thesis:

end;

A47: for p being Point of (TOP-REAL 2) st p in LSeg |[b,c]|,|[a,c]| holds
( 0 <= ((((p `1 ) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1 ) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f3 . (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2)) = p )

proof

let p be Point of (TOP-REAL 2); :: thesis:
assume A48: p in LSeg |[b,c]|,|[a,c]| ; :: thesis:
A49: |[b,c]| `1 = b by EUCLID:56;
A50: |[a,c]| `1 = a by EUCLID:56;
then A51: a <= p `1 by A1, A48, A49, TOPREAL1:9;
A52: p `1 <= b by A1, A48, A49, A50, TOPREAL1:9;
b - a > 0 by A1, XREAL_1:52;
then A53: - (b - a) < - 0 by XREAL_1:26;
b - (p `1 ) >= 0 by A52, XREAL_1:50;
then A54: - (b - (p `1 )) <= - 0 ;
then A55: ((((p `1 ) - b) / (a - b)) / 2) + (1 / 2) >= 0 + (1 / 2) by A53, XREAL_1:9;
(p `1 ) - b >= a - b by A51, XREAL_1:11;
then ((p `1 ) - b) / (a - b) <= (a - b) / (a - b) by A53, XREAL_1:75;
then ((p `1 ) - b) / (a - b) <= 1 by A53, XCMPLX_1:60;
then (((p `1 ) - b) / (a - b)) / 2 <= 1 / 2 by XREAL_1:74;
then A56: ((((p `1 ) - b) / (a - b)) / 2) + (1 / 2) <= (1 / 2) + (1 / 2) by XREAL_1:9;
set r = ((((p `1 ) - b) / (a - b)) / 2) + (1 / 2);
((((p `1 ) - b) / (a - b)) / 2) + (1 / 2) in [.(1 / 2),1.] by A55, A56, XXREAL_1:1;
then f3 . (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2)) = ((1 - ((2 * (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * |[b,c]|) + (((2 * (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) - 1) * |[a,c]|) by A35
.= |[((1 - ((2 * (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * b),((1 - ((2 * (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * c)]| + (((2 * (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) - 1) * |[a,c]|) by EUCLID:62
.= |[((1 - ((2 * (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * b),((1 - ((2 * (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * c)]| + |[(((2 * (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) - 1) * a),(((2 * (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) - 1) * c)]| by EUCLID:62
.= |[(((1 - ((2 * (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) - 1)) * b) + (((2 * (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) - 1) * a)),(((1 * c) - (((2 * (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) - 1) * c)) + (((2 * (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2))) - 1) * c))]| by EUCLID:60
.= |[((1 * b) + ((((p `1 ) - b) / (a - b)) * (a - b))),c]|
.= |[((1 * b) + ((p `1 ) - b)),c]| by A53, XCMPLX_1:88
.= |[(p `1 ),(p `2 )]| by A48, TOPREAL3:18
.= p by EUCLID:57 ;
hence ( 0 <= ((((p `1 ) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1 ) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f3 . (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2)) = p ) by A53, A54, A56; :: thesis:

end;

reconsider B00 = [.0 ,1.] as Subset of R^1 by TOPMETR:24;
reconsider B01 = B00 as non empty Subset of R^1 by XXREAL_1:1;
I[01] = R^1 | B01 by TOPMETR:26, TOPMETR:27;
then consider h1 being Function of I[01] ,R^1 such that
A57: for p being Point of I[01] holds h1 . p = p and
A58: h1 is continuous by Th14;
consider h2 being Function of I[01] ,R^1 such that
A59: for p being Point of I[01]
for r1 being real number st h1 . p = r1 holds
h2 . p = 2 * r1 and
A60: h2 is continuous by A58, JGRAPH_2:33;
consider h5 being Function of I[01] ,R^1 such that
A61: for p being Point of I[01]
for r1 being real number st h2 . p = r1 holds
h5 . p = 1 - r1 and
A62: h5 is continuous by A60, Th16;
consider h3 being Function of I[01] ,R^1 such that
A63: for p being Point of I[01]
for r1 being real number st h2 . p = r1 holds
h3 . p = r1 - 1 and
A64: h3 is continuous by A60, Th15;
consider h4 being Function of I[01] ,R^1 such that
A65: for p being Point of I[01]
for r1 being real number st h3 . p = r1 holds
h4 . p = 1 - r1 and
A66: h4 is continuous by A64, Th16;
consider g1 being Function of I[01] ,(TOP-REAL 2) such that
A67: for r being Point of I[01] holds g1 . r = ((h5 . r) * |[b,d]|) + ((h2 . r) * |[b,c]|) and
A68: g1 is continuous by A60, A62, Th21;
A69: for r being Point of I[01]
for s being real number st r = s holds
g1 . r = ((1 - (2 * s)) * |[b,d]|) + ((2 * s) * |[b,c]|)

proof

let r be Point of I[01] ; :: thesis:
let s be real number ; :: thesis:
assume A70: r = s ; :: thesis:
g1 . r = ((h5 . r) * |[b,d]|) + ((h2 . r) * |[b,c]|) by A67
.= ((1 - (2 * (h1 . r))) * |[b,d]|) + ((h2 . r) * |[b,c]|) by A59, A61
.= ((1 - (2 * (h1 . r))) * |[b,d]|) + ((2 * (h1 . r)) * |[b,c]|) by A59
.= ((1 - (2 * s)) * |[b,d]|) + ((2 * (h1 . r)) * |[b,c]|) by A57, A70
.= ((1 - (2 * s)) * |[b,d]|) + ((2 * s) * |[b,c]|) by A57, A70 ;
hence g1 . r = ((1 - (2 * s)) * |[b,d]|) + ((2 * s) * |[b,c]|) ; :: thesis:

end;

consider g2 being Function of I[01] ,(TOP-REAL 2) such that
A71: for r being Point of I[01] holds g2 . r = ((h4 . r) * |[b,c]|) + ((h3 . r) * |[a,c]|) and
A72: g2 is continuous by A64, A66, Th21;
A73: for r being Point of I[01]
for s being real number st r = s holds
g2 . r = ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * s) - 1) * |[a,c]|)

proof

let r be Point of I[01] ; :: thesis:
let s be real number ; :: thesis:
assume A74: r = s ; :: thesis:
g2 . r = ((h4 . r) * |[b,c]|) + ((h3 . r) * |[a,c]|) by A71
.= ((1 - ((h2 . r) - 1)) * |[b,c]|) + ((h3 . r) * |[a,c]|) by A63, A65
.= ((1 - ((h2 . r) - 1)) * |[b,c]|) + (((h2 . r) - 1) * |[a,c]|) by A63
.= ((1 - ((2 * (h1 . r)) - 1)) * |[b,c]|) + (((h2 . r) - 1) * |[a,c]|) by A59
.= ((1 - ((2 * (h1 . r)) - 1)) * |[b,c]|) + (((2 * (h1 . r)) - 1) * |[a,c]|) by A59
.= ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * (h1 . r)) - 1) * |[a,c]|) by A57, A74
.= ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * s) - 1) * |[a,c]|) by A57, A74 ;
hence g2 . r = ((1 - ((2 * s) - 1)) * |[b,c]|) + (((2 * s) - 1) * |[a,c]|) ; :: thesis:

end;

reconsider B11 = [.0 ,(1 / 2).] as non empty Subset of I[01] by A3, BORSUK_1:83, XBOOLE_1:7, XXREAL_1:1;
A75: dom (g1 | B11) = (dom g1) /\ B11 by RELAT_1:90
.= the carrier of I[01] /\ B11 by FUNCT_2:def 1
.= B11 by XBOOLE_1:28
.= the carrier of (I[01] | B11) by PRE_TOPC:29 ;
rng (g1 | B11) c= the carrier of (TOP-REAL 2) ;
then reconsider g11 = g1 | B11 as Function of (I[01] | B11),(TOP-REAL 2) by A75, FUNCT_2:4;
A76: TOP-REAL 2 is SubSpace of TOP-REAL 2 by TSEP_1:2;
then A77: g11 is continuous by A68, BORSUK_4:69;
reconsider B22 = [.(1 / 2),1.] as non empty Subset of I[01] by A3, BORSUK_1:83, XBOOLE_1:7, XXREAL_1:1;
A78: dom (g2 | B22) = (dom g2) /\ B22 by RELAT_1:90
.= the carrier of I[01] /\ B22 by FUNCT_2:def 1
.= B22 by XBOOLE_1:28
.= the carrier of (I[01] | B22) by PRE_TOPC:29 ;
rng (g2 | B22) c= the carrier of (TOP-REAL 2) ;
then reconsider g22 = g2 | B22 as Function of (I[01] | B22),(TOP-REAL 2) by A78, FUNCT_2:4;
A79: g22 is continuous by A72, A76, BORSUK_4:69;
A80: B11 = [#] (I[01] | B11) by PRE_TOPC:def 10;
A81: B22 = [#] (I[01] | B22) by PRE_TOPC:def 10;
A82: B11 is closed by Th12;
A83: B22 is closed by Th12;
A84: ([#] (I[01] | B11)) \/ ([#] (I[01] | B22)) = [#] I[01] by A80, A81, BORSUK_1:83, XXREAL_1:165;
for p being set st p in ([#] (I[01] | B11)) /\ ([#] (I[01] | B22)) holds
g11 . p = g22 . p

proof

let p be set ; :: thesis:
assume A85: p in ([#] (I[01] | B11)) /\ ([#] (I[01] | B22)) ; :: thesis:
then A86: p in [#] (I[01] | B11) by XBOOLE_0:def 4;
A87: p in [#] (I[01] | B22) by A85;
A88: p in B11 by A86, PRE_TOPC:def 10;
A89: p in B22 by A87, PRE_TOPC:def 10;
reconsider rp = p as Real by A88;
A90: rp <= 1 / 2 by A88, XXREAL_1:1;
rp >= 1 / 2 by A89, XXREAL_1:1;
then rp = 1 / 2 by A90, XXREAL_0:1;
then A91: 2 * rp = 1 ;
thus g11 . p = g1 . p by A88, FUNCT_1:72
.= ((1 - 1) * |[b,d]|) + (1 * |[b,c]|) by A69, A88, A91
.= (0. (TOP-REAL 2)) + (1 * |[b,c]|) by EUCLID:33
.= ((1 - 0 ) * |[b,c]|) + ((1 - 1) * |[a,c]|) by EUCLID:33
.= g2 . p by A73, A88, A91
.= g22 . p by A89, FUNCT_1:72 ; :: thesis:

end;

then consider h being Function of I[01] ,(TOP-REAL 2) such that
A92: h = g11 +* g22 and
A93: h is continuous by A77, A79, A80, A81, A82, A83, A84, JGRAPH_2:9;
A94: dom f3 = dom h by Th13;
A95: dom f3 = the carrier of I[01] by Th13;
for x being set st x in dom f2 holds
f3 . x = h . x

proof

let x be set ; :: thesis:
assume A96: x in dom f2 ; :: thesis:
then reconsider rx = x as Real by A95, BORSUK_1:83;
A97: 0 <= rx by A94, A96, BORSUK_1:83, XXREAL_1:1;
A98: rx <= 1 by A94, A96, BORSUK_1:83, XXREAL_1:1;

per cases ;

suppose A99: rx < 1 / 2 ; :: thesis:

then A100: rx in [.0 ,(1 / 2).] by A97, XXREAL_1:1;
not rx in dom g22 by A81, A99, XXREAL_1:1;
then h . rx = g11 . rx by A92, FUNCT_4:12
.= g1 . rx by A100, FUNCT_1:72
.= ((1 - (2 * rx)) * |[b,d]|) + ((2 * rx) * |[b,c]|) by A69, A94, A96
.= f3 . rx by A32, A100 ;
hence f3 . x = h . x ; :: thesis:

end;

suppose rx >= 1 / 2 ; :: thesis:

then A101: rx in [.(1 / 2),1.] by A98, XXREAL_1:1;
then rx in [#] (I[01] | B22) by PRE_TOPC:def 10;
then h . rx = g22 . rx by A78, A92, FUNCT_4:14
.= g2 . rx by A101, FUNCT_1:72
.= ((1 - ((2 * rx) - 1)) * |[b,c]|) + (((2 * rx) - 1) * |[a,c]|) by A73, A94, A96
.= f3 . rx by A35, A101 ;
hence f3 . x = h . x ; :: thesis:

end;

then A102: f2 = h by A94, FUNCT_1:9;
A103: dom f3 = [#] I[01] by A14, BORSUK_1:83;
for x1, x2 being set st x1 in dom f3 & x2 in dom f3 & f3 . x1 = f3 . x2 holds
x1 = x2

proof

let x1, x2 be set ; :: thesis:
assume that
A104: x1 in dom f3 and
A105: x2 in dom f3 and
A106: f3 . x1 = f3 . x2 ; :: thesis:
reconsider r1 = x1 as Real by A14, A104;
reconsider r2 = x2 as Real by A14, A105;
A107: (LSeg |[b,d]|,|[b,c]|) /\ (LSeg |[b,c]|,|[a,c]|) = {|[b,c]|} by A1, A2, Th42;

now

per cases by A3, A14, A104, A105, XBOOLE_0:def 3;

case A108: ( x1 in [.0 ,(1 / 2).] & x2 in [.0 ,(1 / 2).] ) ; :: thesis:

then f3 . r1 = ((1 - (2 * r1)) * |[b,d]|) + ((2 * r1) * |[b,c]|) by A32;
then ((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|) = ((1 - (2 * r1)) * |[b,d]|) + ((2 * r1) * |[b,c]|) by A32, A106, A108;
then (((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|)) - ((2 * r1) * |[b,c]|) = (1 - (2 * r1)) * |[b,d]| by EUCLID:52;
then ((1 - (2 * r2)) * |[b,d]|) + (((2 * r2) * |[b,c]|) - ((2 * r1) * |[b,c]|)) = (1 - (2 * r1)) * |[b,d]| by EUCLID:49;
then ((1 - (2 * r2)) * |[b,d]|) + (((2 * r2) - (2 * r1)) * |[b,c]|) = (1 - (2 * r1)) * |[b,d]| by EUCLID:54;
then (((2 * r2) - (2 * r1)) * |[b,c]|) + (((1 - (2 * r2)) * |[b,d]|) - ((1 - (2 * r1)) * |[b,d]|)) = ((1 - (2 * r1)) * |[b,d]|) - ((1 - (2 * r1)) * |[b,d]|) by EUCLID:49;
then (((2 * r2) - (2 * r1)) * |[b,c]|) + (((1 - (2 * r2)) * |[b,d]|) - ((1 - (2 * r1)) * |[b,d]|)) = 0. (TOP-REAL 2) by EUCLID:46;
then (((2 * r2) - (2 * r1)) * |[b,c]|) + (((1 - (2 * r2)) - (1 - (2 * r1))) * |[b,d]|) = 0. (TOP-REAL 2) by EUCLID:54;
then (((2 * r2) - (2 * r1)) * |[b,c]|) + ((- ((2 * r2) - (2 * r1))) * |[b,d]|) = 0. (TOP-REAL 2) ;
then (((2 * r2) - (2 * r1)) * |[b,c]|) + (- (((2 * r2) - (2 * r1)) * |[b,d]|)) = 0. (TOP-REAL 2) by EUCLID:44;
then (((2 * r2) - (2 * r1)) * |[b,c]|) - (((2 * r2) - (2 * r1)) * |[b,d]|) = 0. (TOP-REAL 2) by EUCLID:45;
then ((2 * r2) - (2 * r1)) * (|[b,c]| - |[b,d]|) = 0. (TOP-REAL 2) by EUCLID:53;
then ( (2 * r2) - (2 * r1) = 0 or |[b,c]| - |[b,d]| = 0. (TOP-REAL 2) ) by EUCLID:35;
then ( (2 * r2) - (2 * r1) = 0 or |[b,c]| = |[b,d]| ) by EUCLID:47;
then ( (2 * r2) - (2 * r1) = 0 or d = |[b,c]| `2 ) by EUCLID:56;
hence x1 = x2 by A2, EUCLID:56; :: thesis:

end;

case A109: ( x1 in [.0 ,(1 / 2).] & x2 in [.(1 / 2),1.] ) ; :: thesis:

then A110: f3 . r1 = ((1 - (2 * r1)) * |[b,d]|) + ((2 * r1) * |[b,c]|) by A32;
A111: 0 <= r1 by A109, XXREAL_1:1;
r1 <= 1 / 2 by A109, XXREAL_1:1;
then r1 * 2 <= (1 / 2) * 2 by XREAL_1:66;
then A112: f3 . r1 in LSeg |[b,d]|,|[b,c]| by A110, A111;
A113: f3 . r2 = ((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|) by A35, A109;
A114: 1 / 2 <= r2 by A109, XXREAL_1:1;
A115: r2 <= 1 by A109, XXREAL_1:1;
r2 * 2 >= (1 / 2) * 2 by A114, XREAL_1:66;
then A116: (2 * r2) - 1 >= 0 by XREAL_1:50;
2 * 1 >= 2 * r2 by A115, XREAL_1:66;
then (1 + 1) - 1 >= (2 * r2) - 1 by XREAL_1:11;
then f3 . r2 in { (((1 - lambda) * |[b,c]|) + (lambda * |[a,c]|)) where lambda is Real : ( 0 <= lambda & lambda <= 1 ) } by A113, A116;
then f3 . r1 in (LSeg |[b,d]|,|[b,c]|) /\ (LSeg |[b,c]|,|[a,c]|) by A106, A112, XBOOLE_0:def 4;
then A117: f3 . r1 = |[b,c]| by A107, TARSKI:def 1;
then (((1 - (2 * r1)) * |[b,d]|) + ((2 * r1) * |[b,c]|)) + (- |[b,c]|) = 0. (TOP-REAL 2) by A110, EUCLID:40;
then (((1 - (2 * r1)) * |[b,d]|) + ((2 * r1) * |[b,c]|)) + ((- 1) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:43;
then ((1 - (2 * r1)) * |[b,d]|) + (((2 * r1) * |[b,c]|) + ((- 1) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:30;
then ((1 - (2 * r1)) * |[b,d]|) + (((2 * r1) + (- 1)) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:37;
then ((1 - (2 * r1)) * |[b,d]|) + ((- (1 - (2 * r1))) * |[b,c]|) = 0. (TOP-REAL 2) ;
then ((1 - (2 * r1)) * |[b,d]|) + (- ((1 - (2 * r1)) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:44;
then ((1 - (2 * r1)) * |[b,d]|) - ((1 - (2 * r1)) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:45;
then (1 - (2 * r1)) * (|[b,d]| - |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:53;
then ( 1 - (2 * r1) = 0 or |[b,d]| - |[b,c]| = 0. (TOP-REAL 2) ) by EUCLID:35;
then ( 1 - (2 * r1) = 0 or |[b,d]| = |[b,c]| ) by EUCLID:47;
then A118: ( 1 - (2 * r1) = 0 or d = |[b,c]| `2 ) by EUCLID:56;
(((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|)) + (- |[b,c]|) = 0. (TOP-REAL 2) by A106, A113, A117, EUCLID:40;
then (((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|)) + ((- 1) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:43;
then (((2 * r2) - 1) * |[a,c]|) + (((1 - ((2 * r2) - 1)) * |[b,c]|) + ((- 1) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:30;
then (((2 * r2) - 1) * |[a,c]|) + (((1 - ((2 * r2) - 1)) + (- 1)) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:37;
then (((2 * r2) - 1) * |[a,c]|) + ((- ((2 * r2) - 1)) * |[b,c]|) = 0. (TOP-REAL 2) ;
then (((2 * r2) - 1) * |[a,c]|) + (- (((2 * r2) - 1) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:44;
then (((2 * r2) - 1) * |[a,c]|) - (((2 * r2) - 1) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:45;
then ((2 * r2) - 1) * (|[a,c]| - |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:53;
then ( (2 * r2) - 1 = 0 or |[a,c]| - |[b,c]| = 0. (TOP-REAL 2) ) by EUCLID:35;
then ( (2 * r2) - 1 = 0 or |[a,c]| = |[b,c]| ) by EUCLID:47;
then ( (2 * r2) - 1 = 0 or a = |[b,c]| `1 ) by EUCLID:56;
hence x1 = x2 by A1, A2, A118, EUCLID:56; :: thesis:

end;

case A119: ( x1 in [.(1 / 2),1.] & x2 in [.0 ,(1 / 2).] ) ; :: thesis:

then A120: f3 . r2 = ((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|) by A32;
A121: 0 <= r2 by A119, XXREAL_1:1;
r2 <= 1 / 2 by A119, XXREAL_1:1;
then r2 * 2 <= (1 / 2) * 2 by XREAL_1:66;
then A122: f3 . r2 in LSeg |[b,d]|,|[b,c]| by A120, A121;
A123: f3 . r1 = ((1 - ((2 * r1) - 1)) * |[b,c]|) + (((2 * r1) - 1) * |[a,c]|) by A35, A119;
A124: 1 / 2 <= r1 by A119, XXREAL_1:1;
A125: r1 <= 1 by A119, XXREAL_1:1;
r1 * 2 >= (1 / 2) * 2 by A124, XREAL_1:66;
then A126: (2 * r1) - 1 >= 0 by XREAL_1:50;
2 * 1 >= 2 * r1 by A125, XREAL_1:66;
then (1 + 1) - 1 >= (2 * r1) - 1 by XREAL_1:11;
then f3 . r1 in { (((1 - lambda) * |[b,c]|) + (lambda * |[a,c]|)) where lambda is Real : ( 0 <= lambda & lambda <= 1 ) } by A123, A126;
then f3 . r2 in (LSeg |[b,d]|,|[b,c]|) /\ (LSeg |[b,c]|,|[a,c]|) by A106, A122, XBOOLE_0:def 4;
then A127: f3 . r2 = |[b,c]| by A107, TARSKI:def 1;
then (((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|)) + (- |[b,c]|) = 0. (TOP-REAL 2) by A120, EUCLID:40;
then (((1 - (2 * r2)) * |[b,d]|) + ((2 * r2) * |[b,c]|)) + ((- 1) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:43;
then ((1 - (2 * r2)) * |[b,d]|) + (((2 * r2) * |[b,c]|) + ((- 1) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:30;
then ((1 - (2 * r2)) * |[b,d]|) + (((2 * r2) + (- 1)) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:37;
then ((1 - (2 * r2)) * |[b,d]|) + ((- (1 - (2 * r2))) * |[b,c]|) = 0. (TOP-REAL 2) ;
then ((1 - (2 * r2)) * |[b,d]|) + (- ((1 - (2 * r2)) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:44;
then ((1 - (2 * r2)) * |[b,d]|) - ((1 - (2 * r2)) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:45;
then (1 - (2 * r2)) * (|[b,d]| - |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:53;
then ( 1 - (2 * r2) = 0 or |[b,d]| - |[b,c]| = 0. (TOP-REAL 2) ) by EUCLID:35;
then ( 1 - (2 * r2) = 0 or |[b,d]| = |[b,c]| ) by EUCLID:47;
then A128: ( 1 - (2 * r2) = 0 or d = |[b,c]| `2 ) by EUCLID:56;
(((1 - ((2 * r1) - 1)) * |[b,c]|) + (((2 * r1) - 1) * |[a,c]|)) + (- |[b,c]|) = 0. (TOP-REAL 2) by A106, A123, A127, EUCLID:40;
then (((1 - ((2 * r1) - 1)) * |[b,c]|) + (((2 * r1) - 1) * |[a,c]|)) + ((- 1) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:43;
then (((2 * r1) - 1) * |[a,c]|) + (((1 - ((2 * r1) - 1)) * |[b,c]|) + ((- 1) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:30;
then (((2 * r1) - 1) * |[a,c]|) + (((1 - ((2 * r1) - 1)) + (- 1)) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:37;
then (((2 * r1) - 1) * |[a,c]|) + ((- ((2 * r1) - 1)) * |[b,c]|) = 0. (TOP-REAL 2) ;
then (((2 * r1) - 1) * |[a,c]|) + (- (((2 * r1) - 1) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:44;
then (((2 * r1) - 1) * |[a,c]|) - (((2 * r1) - 1) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:45;
then ((2 * r1) - 1) * (|[a,c]| - |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:53;
then ( (2 * r1) - 1 = 0 or |[a,c]| - |[b,c]| = 0. (TOP-REAL 2) ) by EUCLID:35;
then ( (2 * r1) - 1 = 0 or |[a,c]| = |[b,c]| ) by EUCLID:47;
then ( (2 * r1) - 1 = 0 or a = |[b,c]| `1 ) by EUCLID:56;
hence x1 = x2 by A1, A2, A128, EUCLID:56; :: thesis:

end;

case A129: ( x1 in [.(1 / 2),1.] & x2 in [.(1 / 2),1.] ) ; :: thesis:

then f3 . r1 = ((1 - ((2 * r1) - 1)) * |[b,c]|) + (((2 * r1) - 1) * |[a,c]|) by A35;
then ((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|) = ((1 - ((2 * r1) - 1)) * |[b,c]|) + (((2 * r1) - 1) * |[a,c]|) by A35, A106, A129;
then (((1 - ((2 * r2) - 1)) * |[b,c]|) + (((2 * r2) - 1) * |[a,c]|)) - (((2 * r1) - 1) * |[a,c]|) = (1 - ((2 * r1) - 1)) * |[b,c]| by EUCLID:52;
then ((1 - ((2 * r2) - 1)) * |[b,c]|) + ((((2 * r2) - 1) * |[a,c]|) - (((2 * r1) - 1) * |[a,c]|)) = (1 - ((2 * r1) - 1)) * |[b,c]| by EUCLID:49;
then ((1 - ((2 * r2) - 1)) * |[b,c]|) + ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) = (1 - ((2 * r1) - 1)) * |[b,c]| by EUCLID:54;
then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) + (((1 - ((2 * r2) - 1)) * |[b,c]|) - ((1 - ((2 * r1) - 1)) * |[b,c]|)) = ((1 - ((2 * r1) - 1)) * |[b,c]|) - ((1 - ((2 * r1) - 1)) * |[b,c]|) by EUCLID:49;
then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) + (((1 - ((2 * r2) - 1)) * |[b,c]|) - ((1 - ((2 * r1) - 1)) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:46;
then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) + (((1 - ((2 * r2) - 1)) - (1 - ((2 * r1) - 1))) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:54;
then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) + ((- (((2 * r2) - 1) - ((2 * r1) - 1))) * |[b,c]|) = 0. (TOP-REAL 2) ;
then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) + (- ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b,c]|)) = 0. (TOP-REAL 2) by EUCLID:44;
then ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[a,c]|) - ((((2 * r2) - 1) - ((2 * r1) - 1)) * |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:45;
then (((2 * r2) - 1) - ((2 * r1) - 1)) * (|[a,c]| - |[b,c]|) = 0. (TOP-REAL 2) by EUCLID:53;
then ( ((2 * r2) - 1) - ((2 * r1) - 1) = 0 or |[a,c]| - |[b,c]| = 0. (TOP-REAL 2) ) by EUCLID:35;
then ( ((2 * r2) - 1) - ((2 * r1) - 1) = 0 or |[a,c]| = |[b,c]| ) by EUCLID:47;
then ( ((2 * r2) - 1) - ((2 * r1) - 1) = 0 or a = |[b,c]| `1 ) by EUCLID:56;
hence x1 = x2 by A1, EUCLID:56; :: thesis:

end;

hence x1 = x2 ; :: thesis:

end;

then A130: f3 is one-to-one by FUNCT_1:def 8;
[#] ((TOP-REAL 2) | (Lower_Arc (rectangle a,b,c,d))) c= rng f3

proof

let y be set ; :: according to TARSKI:def 3 :: thesis:
assume y in [#] ((TOP-REAL 2) | (Lower_Arc (rectangle a,b,c,d))) ; :: thesis:
then A131: y in Lower_Arc (rectangle a,b,c,d) by PRE_TOPC:def 10;
then reconsider q = y as Point of (TOP-REAL 2) ;
A132: Lower_Arc (rectangle a,b,c,d) = (LSeg |[b,d]|,|[b,c]|) \/ (LSeg |[b,c]|,|[a,c]|) by A1, A2, Th62;

now

per cases by A131, A132, XBOOLE_0:def 3;

case A133: q in LSeg |[b,d]|,|[b,c]| ; :: thesis:

then A134: 0 <= (((q `2 ) - d) / (c - d)) / 2 by A38;
A135: (((q `2 ) - d) / (c - d)) / 2 <= 1 by A38, A133;
A136: f3 . ((((q `2 ) - d) / (c - d)) / 2) = q by A38, A133;
(((q `2 ) - d) / (c - d)) / 2 in [.0 ,1.] by A134, A135, XXREAL_1:1;
hence y in rng f3 by A14, A136, FUNCT_1:def 5; :: thesis:

end;

case A137: q in LSeg |[b,c]|,|[a,c]| ; :: thesis:

then A138: 0 <= ((((q `1 ) - b) / (a - b)) / 2) + (1 / 2) by A47;
A139: ((((q `1 ) - b) / (a - b)) / 2) + (1 / 2) <= 1 by A47, A137;
A140: f3 . (((((q `1 ) - b) / (a - b)) / 2) + (1 / 2)) = q by A47, A137;
((((q `1 ) - b) / (a - b)) / 2) + (1 / 2) in [.0 ,1.] by A138, A139, XXREAL_1:1;
hence y in rng f3 by A14, A140, FUNCT_1:def 5; :: thesis:

end;

hence y in rng f3 ; :: thesis:

end;

then A141: rng f3 = [#] ((TOP-REAL 2) | (Lower_Arc (rectangle a,b,c,d))) by XBOOLE_0:def 10;
I[01] is compact by HEINE:11, TOPMETR:27;
then A142: f3 is being_homeomorphism by A93, A102, A103, A130, A141, COMPTS_1:26, JGRAPH_1:63;
rng f3 = Lower_Arc (rectangle a,b,c,d) by A141, PRE_TOPC:def 10;
hence ex f being Function of I[01] ,((TOP-REAL 2) | (Lower_Arc (rectangle a,b,c,d))) st
( f is being_homeomorphism & f . 0 = E-max (rectangle a,b,c,d) & f . 1 = W-min (rectangle a,b,c,d) & rng f = Lower_Arc (rectangle a,b,c,d) & ( for r being Real st r in [.0 ,(1 / 2).] holds
f . r = ((1 - (2 * r)) * |[b,d]|) + ((2 * r) * |[b,c]|) ) & ( for r being Real st r in [.(1 / 2),1.] holds
f . r = ((1 - ((2 * r) - 1)) * |[b,c]|) + (((2 * r) - 1) * |[a,c]|) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg |[b,d]|,|[b,c]| holds
( 0 <= (((p `2 ) - d) / (c - d)) / 2 & (((p `2 ) - d) / (c - d)) / 2 <= 1 & f . ((((p `2 ) - d) / (c - d)) / 2) = p ) ) & ( for p being Point of (TOP-REAL 2) st p in LSeg |[b,c]|,|[a,c]| holds
( 0 <= ((((p `1 ) - b) / (a - b)) / 2) + (1 / 2) & ((((p `1 ) - b) / (a - b)) / 2) + (1 / 2) <= 1 & f . (((((p `1 ) - b) / (a - b)) / 2) + (1 / 2)) = p ) ) ) by A29, A31, A32, A35, A38, A47, A142; :: thesis: