thus Sq_Circ is Function of (TOP-REAL 2),(TOP-REAL 2) ; :: thesis: ( rng Sq_Circ = the carrier of (TOP-REAL 2) & ( for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ holds
f is being_homeomorphism ) )
A1: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ holds
( rng Sq_Circ = the carrier of (TOP-REAL 2) & f is being_homeomorphism )

proof

let f be Function of (TOP-REAL 2),(TOP-REAL 2); :: thesis: ( f = Sq_Circ implies ( rng Sq_Circ = the carrier of (TOP-REAL 2) & f is being_homeomorphism ) )
assume A2: f = Sq_Circ ; :: thesis: ( rng Sq_Circ = the carrier of (TOP-REAL 2) & f is being_homeomorphism )
reconsider g = f /" as Function of (TOP-REAL 2),(TOP-REAL 2) ;
A3: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def 1;
the carrier of (TOP-REAL 2) c= rng f

proof

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in the carrier of (TOP-REAL 2) or y in rng f )
assume y in the carrier of (TOP-REAL 2) ; :: thesis: y in rng f
then reconsider p2 = y as Point of (TOP-REAL 2) ;
set q = p2;

now :: thesis: ( ( p2 = 0. (TOP-REAL 2) & ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x ) ) or ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) & ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x ) ) or ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) & ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x ) ) )

per cases ( p2 = 0. (TOP-REAL 2) or ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) or ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ;

case p2 = 0. (TOP-REAL 2) ; :: thesis: ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x )

then y = Sq_Circ . p2 by Def1;
hence ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x ) by A2, A3; :: thesis: verum

end;

case A4: ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ; :: thesis: ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x )

set px = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]|;
A5: sqrt (1 + (((p2 `2) / (p2 `1)) ^2)) > 0 by Lm1, SQUARE_1:25;

A6: now :: thesis: ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 0 implies not |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 = 0 )

assume that
A7: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = 0 and
A8: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 = 0 ; :: thesis: contradiction
(p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = 0 by A8, EUCLID:52;
then A9: p2 `2 = 0 by A5, XCMPLX_1:6;
(p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) = 0 by A7, EUCLID:52;
then p2 `1 = 0 by A5, XCMPLX_1:6;
hence contradiction by A4, A9, EUCLID:53, EUCLID:54; :: thesis: verum

end;

A10: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def 1;
A11: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 = (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52;
A12: |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 = (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by EUCLID:52;
then A13: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) = (p2 `2) / (p2 `1) by A11, A5, XCMPLX_1:91;
then A14: (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) = p2 `2 by A12, A5, XCMPLX_1:89;
( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) ) by A4, A5, XREAL_1:64;
then ( ( p2 `2 <= p2 `1 & (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ) ) by A11, A12, A5, XREAL_1:64;
then ( ( (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) & - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) <= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 >= |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1 & |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) ) ) by A11, A5, EUCLID:52, XREAL_1:64;
then A15: Sq_Circ . |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| = |[((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2)))),((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))))]| by A11, A12, A6, Def1, JGRAPH_2:3;
(|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `2) / (|[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| `1)) ^2))) = p2 `1 by A11, A5, A13, XCMPLX_1:89;
hence ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x ) by A15, A14, A10, EUCLID:53; :: thesis: verum

end;

case A16: ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ; :: thesis: ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x )

set px = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]|;
A17: sqrt (1 + (((p2 `1) / (p2 `2)) ^2)) > 0 by Lm1, SQUARE_1:25;

A18: now :: thesis: ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 0 implies not |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 = 0 )

assume that
A19: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = 0 and
|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 = 0 ; :: thesis: contradiction
(p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) = 0 by A19, EUCLID:52;
then p2 `2 = 0 by A17, XCMPLX_1:6;
hence contradiction by A16; :: thesis: verum

end;

A20: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 = (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52;
A21: |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 = (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by EUCLID:52;
then A22: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) = (p2 `1) / (p2 `2) by A20, A17, XCMPLX_1:91;
then A23: (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) = p2 `1 by A21, A17, XCMPLX_1:89;
( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ) by A16, JGRAPH_2:13;
then ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) ) by A17, XREAL_1:64;
then ( ( p2 `1 <= p2 `2 & (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ) ) by A20, A21, A17, XREAL_1:64;
then ( ( (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) & - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) <= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 ) or ( |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 >= |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2 & |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1 <= - (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) ) ) by A20, A17, EUCLID:52, XREAL_1:64;
then A24: Sq_Circ . |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| = |[((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2)))),((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))))]| by A20, A21, A18, Th4, JGRAPH_2:3;
A25: dom Sq_Circ = the carrier of (TOP-REAL 2) by FUNCT_2:def 1;
(|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2) / (sqrt (1 + (((|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `1) / (|[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| `2)) ^2))) = p2 `2 by A20, A17, A22, XCMPLX_1:89;
hence ex x being set st
( x in dom Sq_Circ & y = Sq_Circ . x ) by A24, A23, A25, EUCLID:53; :: thesis: verum

end;

end;

end;

hence y in rng f by A2, FUNCT_1:def 3; :: thesis: verum

end;

then rng f = the carrier of (TOP-REAL 2) ;
then A26: f is onto by FUNCT_2:def 3;
A27: rng f = dom (f ") by A2, FUNCT_1:33
.= dom (f /") by A2, A26, TOPS_2:def 4
.= [#] (TOP-REAL 2) by FUNCT_2:def 1 ;
g = Sq_Circ " by A26, A2, TOPS_2:def 4;
hence ( rng Sq_Circ = the carrier of (TOP-REAL 2) & f is being_homeomorphism ) by A2, A3, A27, Th21, Th42, TOPS_2:def 5; :: thesis: verum

end;

hence rng Sq_Circ = the carrier of (TOP-REAL 2) ; :: thesis: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ holds
f is being_homeomorphism
thus for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ holds
f is being_homeomorphism by A1; :: thesis: verum