let x, y be real number ; :: thesis: ( 1 <= x & 1 <= y implies (cosh1" x) + (cosh1" y) = cosh1" ((x * y) + (sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1)))) )
assume A1: ( 1 <= x & 1 <= y ) ; :: thesis: (cosh1" x) + (cosh1" y) = cosh1" ((x * y) + (sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1))))
then A2: ( (x ^2 ) - 1 >= 0 & (y ^2 ) - 1 >= 0 ) by Lm4;
then A3: ((x ^2 ) - 1) * ((y ^2 ) - 1) >= 0 * ((y ^2 ) - 1) ;
( 0 < x + (sqrt ((x ^2 ) - 1)) & 0 < y + (sqrt ((y ^2 ) - 1)) ) by A1, Th23;
then A4: (cosh1" x) + (cosh1" y) = log number_e ,((x + (sqrt ((x ^2 ) - 1))) * (y + (sqrt ((y ^2 ) - 1)))) by Lm2, POWER:61
.= log number_e ,((((x * (sqrt ((y ^2 ) - 1))) + (y * (sqrt ((x ^2 ) - 1)))) + ((sqrt ((x ^2 ) - 1)) * (sqrt ((y ^2 ) - 1)))) + (x * y))
.= log number_e ,((((x * (sqrt ((y ^2 ) - 1))) + (y * (sqrt ((x ^2 ) - 1)))) + (sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1)))) + (x * y)) by A2, SQUARE_1:97 ;
set t = (x * (sqrt ((y ^2 ) - 1))) + (y * (sqrt ((x ^2 ) - 1)));
(x * (sqrt ((y ^2 ) - 1))) + (y * (sqrt ((x ^2 ) - 1))) = sqrt (((x * (sqrt ((y ^2 ) - 1))) + (y * (sqrt ((x ^2 ) - 1)))) ^2 ) by A1, Th24, SQUARE_1:89
.= sqrt ((((x ^2 ) * ((sqrt ((y ^2 ) - 1)) ^2 )) + ((2 * (x * (sqrt ((y ^2 ) - 1)))) * (y * (sqrt ((x ^2 ) - 1))))) + ((y * (sqrt ((x ^2 ) - 1))) ^2 ))
.= sqrt ((((x ^2 ) * ((y ^2 ) - 1)) + ((2 * (x * (sqrt ((y ^2 ) - 1)))) * (y * (sqrt ((x ^2 ) - 1))))) + ((y * (sqrt ((x ^2 ) - 1))) ^2 )) by A2, SQUARE_1:def 4
.= sqrt (((((x ^2 ) * (y ^2 )) - (x ^2 )) + ((2 * (x * (sqrt ((y ^2 ) - 1)))) * (y * (sqrt ((x ^2 ) - 1))))) + ((y ^2 ) * ((sqrt ((x ^2 ) - 1)) ^2 )))
.= sqrt (((((x ^2 ) * (y ^2 )) - (x ^2 )) + ((2 * (x * (sqrt ((y ^2 ) - 1)))) * (y * (sqrt ((x ^2 ) - 1))))) + ((y ^2 ) * ((x ^2 ) - 1))) by A2, SQUARE_1:def 4
.= sqrt ((((2 * ((x * y) ^2 )) - (x ^2 )) - (y ^2 )) + ((2 * (x * (sqrt ((y ^2 ) - 1)))) * (y * (sqrt ((x ^2 ) - 1))))) ;
then A5: log number_e ,((((x * (sqrt ((y ^2 ) - 1))) + (y * (sqrt ((x ^2 ) - 1)))) + (sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1)))) + (x * y)) = log number_e ,(((sqrt ((((2 * ((x * y) ^2 )) - (x ^2 )) - (y ^2 )) + (((2 * x) * y) * ((sqrt ((y ^2 ) - 1)) * (sqrt ((x ^2 ) - 1)))))) + (sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1)))) + (x * y))
.= log number_e ,(((sqrt ((((2 * ((x * y) ^2 )) - (x ^2 )) - (y ^2 )) + (((2 * x) * y) * (sqrt (((y ^2 ) - 1) * ((x ^2 ) - 1)))))) + (sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1)))) + (x * y)) by A2, SQUARE_1:97 ;
cosh1" ((x * y) + (sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1)))) = log number_e ,(((x * y) + (sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1)))) + (sqrt (((((x * y) ^2 ) + ((2 * (x * y)) * (sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1))))) + ((sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1))) ^2 )) - 1)))
.= log number_e ,(((x * y) + (sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1)))) + (sqrt (((((x * y) ^2 ) + ((2 * (x * y)) * (sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1))))) + (((x ^2 ) - 1) * ((y ^2 ) - 1))) - 1))) by A3, SQUARE_1:def 4
.= log number_e ,((sqrt ((((2 * ((x * y) ^2 )) - (x ^2 )) - (y ^2 )) + (((2 * x) * y) * (sqrt (((y ^2 ) - 1) * ((x ^2 ) - 1)))))) + ((sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1))) + (x * y))) ;
hence (cosh1" x) + (cosh1" y) = cosh1" ((x * y) + (sqrt (((x ^2 ) - 1) * ((y ^2 ) - 1)))) by A4, A5; :: thesis: verum