let F, G be Function of V1,V2; :: thesis: ( ( for v being Element of V1 holds F . v = a * (f . v) ) & ( for v being Element of V1 holds G . v = a * (f . v) ) implies F = G )

assume that

A2: for v being Element of V1 holds F . v = a * (f . v) and

A3: for v being Element of V1 holds G . v = a * (f . v) ; :: thesis: F = G

assume that

A2: for v being Element of V1 holds F . v = a * (f . v) and

A3: for v being Element of V1 holds G . v = a * (f . v) ; :: thesis: F = G

now :: thesis: for v being Element of V1 holds F . v = G . v

hence
F = G
by FUNCT_2:63; :: thesis: verumlet v be Element of V1; :: thesis: F . v = G . v

thus F . v = a * (f . v) by A2

.= G . v by A3 ; :: thesis: verum

end;thus F . v = a * (f . v) by A2

.= G . v by A3 ; :: thesis: verum