let f, g be Function of REAL,REAL; :: thesis: ( ( for r being real number holds f . r = abs r ) & ( for r being real number holds g . r = abs r ) implies f = g )
assume that
A2: for r being real number holds f . r = abs r and
A3: for r being real number holds g . r = abs r ; :: thesis: f = g

now

let x be Element of REAL ; :: thesis: f . x = g . x
thus f . x = abs x by A2
.= g . x by A3 ; :: thesis: verum

end;

hence f = g by FUNCT_2:63; :: thesis: verum