defpred S1[ set , set ] means ex x, y being Real st
( \$1 = |[x,y]| & ( y >= 0 implies \$2 = () / (2 * PI) ) & ( y <= 0 implies \$2 = 1 - (() / (2 * PI)) ) );
reconsider A = R^1 ].0,1.[ as non empty Subset of R^1 ;
A1: the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) = A by PRE_TOPC:8;
A2: for x being Element of the carrier of ex y being Element of the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) st S1[x,y]
proof
let x be Element of the carrier of ; :: thesis: ex y being Element of the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) st S1[x,y]
A3: the carrier of = the carrier of () \ by Def10;
then A4: x in the carrier of () by XBOOLE_0:def 5;
A5: the carrier of () is Subset of () by TSEP_1:1;
then consider a, b being Element of REAL such that
A6: x = <*a,b*> by ;
reconsider x1 = x as Point of () by A4, A5;
A7: b = x1 `2 by ;
set k = arccos a;
A8: a = x1 `1 by ;
then A9: - 1 <= a by Th26;
A10: 1 ^2 = |.x1.| ^2 by
.= (a ^2) + (b ^2) by ;
A11: a <= 1 by ;
then A12: 0 <= arccos a by ;
A13: (arccos a) / (2 * PI) <= 1 / 2 by ;
A14: not x in by ;
A15: now :: thesis: not arccos a = 0
assume A16: arccos a = 0 ; :: thesis: contradiction
then 1 - 1 = (1 + (b ^2)) - 1 by ;
then A17: b = 0 ;
a = 1 by ;
hence contradiction by A6, A14, A17, TARSKI:def 1; :: thesis: verum
end;
A18: arccos a <= PI by ;
A19: 0 <= () / (2 * PI) by ;
per cases ( b = 0 or b > 0 or b < 0 ) ;
suppose A20: b = 0 ; :: thesis: ex y being Element of the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) st S1[x,y]
set y = () / (2 * PI);
(arccos a) / (2 * PI) < 1 by ;
then reconsider y = () / (2 * PI) as Element of the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) by ;
take y ; :: thesis: S1[x,y]
take a ; :: thesis: ex y being Real st
( x = |[a,y]| & ( y >= 0 implies y = () / (2 * PI) ) & ( y <= 0 implies y = 1 - (() / (2 * PI)) ) )

take b ; :: thesis: ( x = |[a,b]| & ( b >= 0 implies y = () / (2 * PI) ) & ( b <= 0 implies y = 1 - (() / (2 * PI)) ) )
thus x = |[a,b]| by A6; :: thesis: ( ( b >= 0 implies y = () / (2 * PI) ) & ( b <= 0 implies y = 1 - (() / (2 * PI)) ) )
thus ( b >= 0 implies y = () / (2 * PI) ) ; :: thesis: ( b <= 0 implies y = 1 - (() / (2 * PI)) )
assume b <= 0 ; :: thesis: y = 1 - (() / (2 * PI))
A21: a <> 1 by ;
hence y = (1 * PI) / (2 * PI) by
.= 1 - (1 / 2) by XCMPLX_1:91
.= 1 - ((1 * PI) / (2 * PI)) by XCMPLX_1:91
.= 1 - (() / (2 * PI)) by ;
:: thesis: verum
end;
suppose A22: b > 0 ; :: thesis: ex y being Element of the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) st S1[x,y]
set y = () / (2 * PI);
(arccos a) / (2 * PI) < 1 by ;
then reconsider y = () / (2 * PI) as Element of the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) by ;
take y ; :: thesis: S1[x,y]
take a ; :: thesis: ex y being Real st
( x = |[a,y]| & ( y >= 0 implies y = () / (2 * PI) ) & ( y <= 0 implies y = 1 - (() / (2 * PI)) ) )

take b ; :: thesis: ( x = |[a,b]| & ( b >= 0 implies y = () / (2 * PI) ) & ( b <= 0 implies y = 1 - (() / (2 * PI)) ) )
thus ( x = |[a,b]| & ( b >= 0 implies y = () / (2 * PI) ) & ( b <= 0 implies y = 1 - (() / (2 * PI)) ) ) by ; :: thesis: verum
end;
suppose A23: b < 0 ; :: thesis: ex y being Element of the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) st S1[x,y]
set y = 1 - (() / (2 * PI));
A24: ((2 * PI) - ()) / (2 * PI) = ((2 * PI) / (2 * PI)) - (() / (2 * PI)) by XCMPLX_1:120
.= 1 - (() / (2 * PI)) by XCMPLX_1:60 ;
(2 * PI) - () < (2 * PI) - 0 by ;
then 1 - (() / (2 * PI)) < (2 * PI) / (2 * PI) by ;
then A25: 1 - (() / (2 * PI)) < 1 by XCMPLX_1:60;
1 * () < 2 * PI by ;
then (arccos a) - () < (2 * PI) - () by XREAL_1:14;
then reconsider y = 1 - (() / (2 * PI)) as Element of the carrier of (R^1 | (R^1 ].0,(0 + p1).[)) by ;
take y ; :: thesis: S1[x,y]
take a ; :: thesis: ex y being Real st
( x = |[a,y]| & ( y >= 0 implies y = () / (2 * PI) ) & ( y <= 0 implies y = 1 - (() / (2 * PI)) ) )

take b ; :: thesis: ( x = |[a,b]| & ( b >= 0 implies y = () / (2 * PI) ) & ( b <= 0 implies y = 1 - (() / (2 * PI)) ) )
thus ( x = |[a,b]| & ( b >= 0 implies y = () / (2 * PI) ) & ( b <= 0 implies y = 1 - (() / (2 * PI)) ) ) by ; :: thesis: verum
end;
end;
end;
ex G being Function of ,(R^1 | (R^1 ].0,(0 + p1).[)) st
for p being Point of holds S1[p,G . p] from hence ex b1 being Function of ,(R^1 | (R^1 ].0,1.[)) st
for p being Point of ex x, y being Real st
( p = |[x,y]| & ( y >= 0 implies b1 . p = () / (2 * PI) ) & ( y <= 0 implies b1 . p = 1 - (() / (2 * PI)) ) ) ; :: thesis: verum