let z be complex number ; :: thesis: Re z >= - |.z.|
0 <= (Im z) ^2 by XREAL_1:65;
then ((Re z) ^2 ) + 0 <= ((Re z) ^2 ) + ((Im z) ^2 ) by XREAL_1:9;
then ( 0 <= (Re z) ^2 & (Re z) ^2 <= ((Re z) ^2 ) + ((Im z) ^2 ) ) by XREAL_1:65;
then sqrt ((Re z) ^2 ) <= |.z.| by SQUARE_1:94;
then - (sqrt ((Re z) ^2 )) >= - |.z.| by XREAL_1:26;
then ( Re z >= - (abs (Re z)) & - (abs (Re z)) >= - |.z.| ) by ABSVALUE:11, COMPLEX1:158;
hence Re z >= - |.z.| by XXREAL_0:2; :: thesis: verum