let x0, x1 be Real; :: thesis: for f being Function of REAL,REAL st ( for x being Real holds f . x = sqrt x ) & x0 <> x1 & x0 > 0 & x1 > 0 holds
[!f,x0,x1!] = 1 / ((sqrt x0) + (sqrt x1))
let f be Function of REAL,REAL; :: thesis: ( ( for x being Real holds f . x = sqrt x ) & x0 <> x1 & x0 > 0 & x1 > 0 implies [!f,x0,x1!] = 1 / ((sqrt x0) + (sqrt x1)) )
assume that
A1: for x being Real holds f . x = sqrt x and
A2: x0 <> x1 and
A3: ( x0 > 0 & x1 > 0 ) ; :: thesis: [!f,x0,x1!] = 1 / ((sqrt x0) + (sqrt x1))
[!f,x0,x1!] = ((sqrt x0) - (f . x1)) / (x0 - x1) by A1
.= ((sqrt x0) - (sqrt x1)) / (x0 - x1) by A1
.= 1 / ((sqrt x0) + (sqrt x1)) by A2, A3, SQUARE_1:36 ;
hence [!f,x0,x1!] = 1 / ((sqrt x0) + (sqrt x1)) ; :: thesis: verum