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

now :: thesis: for x being Element of REAL holds f . x = g . x

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

end;

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