defpred S₁[ set , set ] means ex x, y being Real st
( $1 = |[x,y]| & ( y >= 0 implies $2 = 1 + ((arccos x) / (2 * PI)) ) & ( y <= 0 implies $2 = 1 - ((arccos x) / (2 * PI)) ) );
reconsider A1 = R^1 ].(1 / 2),((1 / 2) + p1).[ as non empty Subset of R^1 ;
A1: the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) = A1 by PRE_TOPC:8;
A2: for x being Element of the carrier of (Topen_unit_circle c[-10]) ex y being Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) st S₁[x,y]

proof

let x be Element of the carrier of (Topen_unit_circle c[-10]); :: thesis: ex y being Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) st S₁[x,y]
A3: the carrier of (Topen_unit_circle c[-10]) = the carrier of (Tunit_circle 2) \ {c[-10]} by Def10;
then A4: x in the carrier of (Tunit_circle 2) by XBOOLE_0:def 5;
A5: not x in {c[-10]} by A3, XBOOLE_0:def 5;
A6: the carrier of (Tunit_circle 2) is Subset of (TOP-REAL 2) by TSEP_1:1;
then consider a, b being Element of REAL such that
A7: x = <*a,b*> by A4, EUCLID:51;
reconsider x1 = x as Point of (TOP-REAL 2) by A4, A6;
A8: b = x1 `2 by A7, EUCLID:52;
set k = arccos a;
A9: a = x1 `1 by A7, EUCLID:52;
then A10: - 1 <= a by Th26;
A11: a <= 1 by A9, Th26;
then A12: (arccos a) / (2 * PI) <= 1 / 2 by A10, Lm22;
A13: 1 ^2 = |.x1.| ^2 by A4, Th12
.= (a ^2) + (b ^2) by A9, A8, JGRAPH_3:1 ;

A14: now :: thesis: not arccos a = PI

assume A15: arccos a = PI ; :: thesis: contradiction
then 1 - 1 = (((- 1) ^2) + (b ^2)) - 1 by A10, A11, A13, SIN_COS6:98
.= ((1 ^2) + (b ^2)) - 1 ;
then A16: b = 0 ;
a = - 1 by A10, A11, A15, SIN_COS6:98;
hence contradiction by A7, A5, A16, TARSKI:def 1; :: thesis: verum

end;

A17: now :: thesis: not (arccos a) / (2 * PI) = 1 / 2

assume (arccos a) / (2 * PI) = 1 / 2 ; :: thesis: contradiction
then ((arccos a) / (2 * PI)) * 2 = (1 / 2) * 2 ;
then (arccos a) / PI = 1 by XCMPLX_1:92;
hence contradiction by A14, XCMPLX_1:58; :: thesis: verum

end;

A18: 0 <= (arccos a) / (2 * PI) by A10, A11, Lm22;

A19: now :: thesis: for y being Real st y = 1 + ((arccos a) / (2 * PI)) holds
y is Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[))

let y be Real; :: thesis: ( y = 1 + ((arccos a) / (2 * PI)) implies y is Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) )
assume A20: y = 1 + ((arccos a) / (2 * PI)) ; :: thesis: y is Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[))
then A21: y <> 1 + (1 / 2) by A17;
1 + 0 <= y by A18, A20, XREAL_1:6;
then A22: 1 / 2 < y by XXREAL_0:2;
y <= 1 + (1 / 2) by A12, A20, XREAL_1:6;
then y < 3 / 2 by A21, XXREAL_0:1;
hence y is Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by A1, A22, XXREAL_1:4; :: thesis: verum

end;

per cases ( b = 0 or b > 0 or b < 0 ) ;

suppose A23: b = 0 ; :: thesis: ex y being Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) st S₁[x,y]

reconsider y = 1 + ((arccos a) / (2 * PI)) as Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by A19;
take y ; :: thesis: S₁[x,y]
take a ; :: thesis: ex y being Real st
( x = |[a,y]| & ( y >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( y <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
take b ; :: thesis: ( x = |[a,b]| & ( b >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
thus x = |[a,b]| by A7; :: thesis: ( ( b >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
a <> - 1 by A7, A5, A23, TARSKI:def 1;
then a = 1 by A13, A23, SQUARE_1:41;
hence ( ( b >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) ) by SIN_COS6:95; :: thesis: verum

end;

suppose A24: b > 0 ; :: thesis: ex y being Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) st S₁[x,y]

reconsider y = 1 + ((arccos a) / (2 * PI)) as Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by A19;
take y ; :: thesis: S₁[x,y]
take a ; :: thesis: ex y being Real st
( x = |[a,y]| & ( y >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( y <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
take b ; :: thesis: ( x = |[a,b]| & ( b >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
thus ( x = |[a,b]| & ( b >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) ) by A7, A24; :: thesis: verum

end;

suppose A25: b < 0 ; :: thesis: ex y being Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) st S₁[x,y]

set y = 1 - ((arccos a) / (2 * PI));
A26: 1 - ((arccos a) / (2 * PI)) <> 1 / 2 by A17;
1 - ((arccos a) / (2 * PI)) >= 1 - (1 / 2) by A12, XREAL_1:13;
then A27: 1 / 2 < 1 - ((arccos a) / (2 * PI)) by A26, XXREAL_0:1;
1 - 0 >= 1 - ((arccos a) / (2 * PI)) by A18, XREAL_1:13;
then 1 - ((arccos a) / (2 * PI)) < 3 / 2 by XXREAL_0:2;
then reconsider y = 1 - ((arccos a) / (2 * PI)) as Element of the carrier of (R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) by A1, A27, XXREAL_1:4;
take y ; :: thesis: S₁[x,y]
take a ; :: thesis: ex y being Real st
( x = |[a,y]| & ( y >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( y <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
take b ; :: thesis: ( x = |[a,b]| & ( b >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) )
thus ( x = |[a,b]| & ( b >= 0 implies y = 1 + ((arccos a) / (2 * PI)) ) & ( b <= 0 implies y = 1 - ((arccos a) / (2 * PI)) ) ) by A7, A25; :: thesis: verum

end;

end;

end;

ex G being Function of (Topen_unit_circle c[-10]),(R^1 | (R^1 ].(1 / 2),((1 / 2) + p1).[)) st
for p being Point of (Topen_unit_circle c[-10]) holds S₁[p,G . p] from FUNCT_2:sch 3(A2);
hence ex b₁ being Function of (Topen_unit_circle c[-10]),(R^1 | (R^1 ].(1 / 2),(3 / 2).[)) st
for p being Point of (Topen_unit_circle c[-10]) ex x, y being Real st
( p = |[x,y]| & ( y >= 0 implies b₁ . p = 1 + ((arccos x) / (2 * PI)) ) & ( y <= 0 implies b₁ . p = 1 - ((arccos x) / (2 * PI)) ) ) ; :: thesis: verum