let a, b be real number ; :: thesis: ( b <= 0 & a <= b implies a * (sqrt (1 + (b ^2 ))) <= b * (sqrt (1 + (a ^2 ))) )
assume that
A1: b <= 0 and
A2: a <= b ; :: thesis: a * (sqrt (1 + (b ^2 ))) <= b * (sqrt (1 + (a ^2 )))
A3: a ^2 >= 0 by XREAL_1:65;
then A4: (- b) * (sqrt (1 + (a ^2 ))) = sqrt (((- b) ^2 ) * (1 + (a ^2 ))) by A1, SQUARE_1:124;
A5: (- a) * (sqrt (1 + (b ^2 ))) = sqrt (((- a) ^2 ) * (1 + (b ^2 ))) by A1, A2, SQUARE_1:124;
( a < b or a = b ) by A2, XXREAL_0:1;
then ( b ^2 < a ^2 or a = b ) by A1, SQUARE_1:114;
then A6: ((b ^2 ) * 1) + ((b ^2 ) * (a ^2 )) <= ((a ^2 ) * 1) + ((a ^2 ) * (b ^2 )) by XREAL_1:9;
b ^2 >= 0 by XREAL_1:65;
then - (a * (sqrt (1 + (b ^2 )))) >= - (b * (sqrt (1 + (a ^2 )))) by A3, A4, A5, A6, SQUARE_1:94;
hence a * (sqrt (1 + (b ^2 ))) <= b * (sqrt (1 + (a ^2 ))) by XREAL_1:26; :: thesis: verum