let p be Point of (); :: thesis: for f being Function of (),() st f = Sq_Circ & p `1 = - 1 & p `2 < 0 holds
( (f . p) `1 < 0 & (f . p) `2 < 0 )

let f be Function of (),(); :: thesis: ( f = Sq_Circ & p `1 = - 1 & p `2 < 0 implies ( (f . p) `1 < 0 & (f . p) `2 < 0 ) )
assume that
A1: f = Sq_Circ and
A2: p `1 = - 1 and
A3: p `2 < 0 ; :: thesis: ( (f . p) `1 < 0 & (f . p) `2 < 0 )
now :: thesis: ( ( p = 0. () & contradiction ) or ( p <> 0. () & (f . p) `1 < 0 & (f . p) `2 < 0 ) )
per cases ( p = 0. () or p <> 0. () ) ;
case A4: p <> 0. () ; :: thesis: ( (f . p) `1 < 0 & (f . p) `2 < 0 )
now :: thesis: ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & (f . p) `1 < 0 & (f . p) `2 < 0 ) or ( not ( p `2 <= p `1 & - (p `1) <= p `2 ) & not ( p `2 >= p `1 & p `2 <= - (p `1) ) & (f . p) `1 < 0 & (f . p) `2 < 0 ) )
per cases ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or ( not ( p `2 <= p `1 & - (p `1) <= p `2 ) & not ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ) ;
case ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; :: thesis: ( (f . p) `1 < 0 & (f . p) `2 < 0 )
then A5: f . p = |[((p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2)))),((p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))))]| by ;
then A6: (f . p) `1 = (p `1) / (sqrt (1 + (((p `2) / (p `1)) ^2))) by EUCLID:52;
(f . p) `2 = (p `2) / (sqrt (1 + (((p `2) / (p `1)) ^2))) by ;
hence ( (f . p) `1 < 0 & (f . p) `2 < 0 ) by ; :: thesis: verum
end;
case ( not ( p `2 <= p `1 & - (p `1) <= p `2 ) & not ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; :: thesis: ( (f . p) `1 < 0 & (f . p) `2 < 0 )
then A7: f . p = |[((p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2)))),((p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))))]| by ;
then A8: (f . p) `1 = (p `1) / (sqrt (1 + (((p `1) / (p `2)) ^2))) by EUCLID:52;
(f . p) `2 = (p `2) / (sqrt (1 + (((p `1) / (p `2)) ^2))) by ;
hence ( (f . p) `1 < 0 & (f . p) `2 < 0 ) by ; :: thesis: verum
end;
end;
end;
hence ( (f . p) `1 < 0 & (f . p) `2 < 0 ) ; :: thesis: verum
end;
end;
end;
hence ( (f . p) `1 < 0 & (f . p) `2 < 0 ) ; :: thesis: verum