let f1, f2 be Function of REAL,REAL; :: thesis: ( ( for d being Real holds f1 . d = ((exp_R . d) - (exp_R . (- d))) / 2 ) & ( for d being Real holds f2 . d = ((exp_R . d) - (exp_R . (- d))) / 2 ) implies f1 = f2 )
assume A2: for d being Real holds f1 . d = ((exp_R . d) - (exp_R . (- d))) / 2 ; :: thesis: ( ex d being Real st not f2 . d = ((exp_R . d) - (exp_R . (- d))) / 2 or f1 = f2 )
assume A3: for d being Real holds f2 . d = ((exp_R . d) - (exp_R . (- d))) / 2 ; :: thesis: f1 = f2
for d being Element of REAL holds f1 . d = f2 . d

proof

let d be Element of REAL ; :: thesis: f1 . d = f2 . d
thus f1 . d = ((exp_R . d) - (exp_R . (- d))) / 2 by A2
.= f2 . d by A3 ; :: thesis: verum

end;

hence f1 = f2 by FUNCT_2:63; :: thesis: verum