let f, g be Function of (REAL n),REAL; :: thesis: ( ( for x being Element of REAL n holds f . x = |.x.| ) & ( for x being Element of REAL n holds g . x = |.x.| ) implies f = g )

assume that

A2: for x being Element of REAL n holds f . x = |.x.| and

A3: for x being Element of REAL n holds g . x = |.x.| ; :: thesis: f = g

assume that

A2: for x being Element of REAL n holds f . x = |.x.| and

A3: for x being Element of REAL n holds g . x = |.x.| ; :: thesis: f = g

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

hence
f = g
by FUNCT_2:63; :: thesis: verumlet x be Element of REAL n; :: thesis: f . x = g . x

thus f . x = |.x.| by A2

.= g . x by A3 ; :: thesis: verum

end;thus f . x = |.x.| by A2

.= g . x by A3 ; :: thesis: verum