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, Th38; :: thesis: verum
end;
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 ))))]|
then ( p `1 <> 0 & - (p `1 ) = p `2 ) by A2, EUCLID:57, EUCLID:58;
then ( (p `1 ) / (p `2 ) = - 1 & (p `2 ) / (p `1 ) = - 1 ) by XCMPLX_1:198, XCMPLX_1:199;
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, Th38; :: 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 Th38; :: thesis: verum
end;
case A7: ( 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 A8: ( ( 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 ))))]|
A9: now
per cases ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) by A8;
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 A7, 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 A7, XXREAL_0:1; :: thesis: verum
end;
end;
end;
now
per cases ( p `1 = p `2 or p `1 = - (p `2 ) ) by A9;
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, A8, Th38; :: thesis: verum
end;
case A10: 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 A11: (p `2 ) / (p `1 ) = - 1 by XCMPLX_1:198;
p `2 <> 0 by A2, A10, EUCLID:57, EUCLID:58;
then (p `1 ) / (p `2 ) = - 1 by A10, 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, A8, A11, Th38; :: 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 Th38; :: 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;
A12: ( - (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, A12, Th38; :: thesis: verum