let p be Point of (TOP-REAL 2); :: thesis: ( p <> 0. (TOP-REAL 2) implies ( ( ( ( p `1 <= p `2 & - (p `2 ) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2 ) ) ) implies Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| ) & ( ( p `1 <= p `2 & - (p `2 ) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2 ) ) or Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| ) ) )
A1: ( - (p `2 ) < p `1 implies - (- (p `2 )) > - (p `1 ) ) by XREAL_1:26;
assume A2: p <> 0. (TOP-REAL 2) ; :: thesis: ( ( ( ( p `1 <= p `2 & - (p `2 ) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2 ) ) ) implies Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| ) & ( ( p `1 <= p `2 & - (p `2 ) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2 ) ) or Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| ) )
hereby :: thesis: ( ( p `1 <= p `2 & - (p `2 ) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2 ) ) or Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| )
assume A3: ( ( p `1 <= p `2 & - (p `2 ) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2 ) ) ) ; :: thesis: Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|
now
per cases ( ( p `1 <= p `2 & - (p `2 ) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2 ) ) ) by A3;
case A4: ( p `1 <= p `2 & - (p `2 ) <= p `1 ) ; :: thesis: Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|
now
assume A5: ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) ; :: thesis: Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|
A6: now
per cases ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) by A5;
case ( p `2 <= p `1 & - (p `1 ) <= p `2 ) ; :: thesis: ( p `1 = p `2 or p `1 = - (p `2 ) )
hence ( p `1 = p `2 or p `1 = - (p `2 ) ) by A4, XXREAL_0:1; :: thesis: verum
end;
case ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ; :: thesis: ( p `1 = p `2 or p `1 = - (p `2 ) )
then - (p `2 ) >= - (- (p `1 )) by XREAL_1:26;
hence ( p `1 = p `2 or p `1 = - (p `2 ) ) by A4, XXREAL_0:1; :: thesis: verum
end;
end;
end;
now
per cases ( p `1 = p `2 or p `1 = - (p `2 ) ) by A6;
case p `1 = p `2 ; :: thesis: Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|
hence Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A2, A5, Def1; :: thesis: verum
end;
case A7: p `1 = - (p `2 ) ; :: thesis: Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|
then ( p `1 <> 0 & - (p `1 ) = p `2 ) by A2, EUCLID:57, EUCLID:58;
then A8: (p `2 ) / (p `1 ) = - 1 by XCMPLX_1:198;
p `2 <> 0 by A2, A7, EUCLID:57, EUCLID:58;
then (p `1 ) / (p `2 ) = - 1 by A7, XCMPLX_1:198;
hence Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A2, A5, A8, Def1; :: thesis: verum
end;
end;
end;
hence Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| ; :: thesis: verum
end;
hence Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A2, Def1; :: thesis: verum
end;
case A9: ( p `1 >= p `2 & p `1 <= - (p `2 ) ) ; :: thesis: Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|
now
assume A10: ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) ; :: thesis: Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|
A11: now
per cases ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) by A10;
case ( p `2 <= p `1 & - (p `1 ) <= p `2 ) ; :: thesis: ( p `1 = p `2 or p `1 = - (p `2 ) )
then - (- (p `1 )) >= - (p `2 ) by XREAL_1:26;
hence ( p `1 = p `2 or p `1 = - (p `2 ) ) by A9, XXREAL_0:1; :: thesis: verum
end;
case ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ; :: thesis: ( p `1 = p `2 or p `1 = - (p `2 ) )
hence ( p `1 = p `2 or p `1 = - (p `2 ) ) by A9, XXREAL_0:1; :: thesis: verum
end;
end;
end;
now
per cases ( p `1 = p `2 or p `1 = - (p `2 ) ) by A11;
case p `1 = p `2 ; :: thesis: Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|
hence Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A2, A10, Def1; :: thesis: verum
end;
case A12: p `1 = - (p `2 ) ; :: thesis: Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]|
then ( p `1 <> 0 & - (p `1 ) = p `2 ) by A2, EUCLID:57, EUCLID:58;
then A13: (p `2 ) / (p `1 ) = - 1 by XCMPLX_1:198;
p `2 <> 0 by A2, A12, EUCLID:57, EUCLID:58;
then (p `1 ) / (p `2 ) = - 1 by A12, XCMPLX_1:198;
hence Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A2, A10, A13, Def1; :: thesis: verum
end;
end;
end;
hence Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| ; :: thesis: verum
end;
hence Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| by A2, Def1; :: thesis: verum
end;
end;
end;
hence Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| ; :: thesis: verum
end;
A14: ( - (p `2 ) > p `1 implies - (- (p `2 )) < - (p `1 ) ) by XREAL_1:26;
assume ( not ( p `1 <= p `2 & - (p `2 ) <= p `1 ) & not ( p `1 >= p `2 & p `1 <= - (p `2 ) ) ) ; :: thesis: Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]|
hence Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| by A2, A1, A14, Def1; :: thesis: verum