let c1, c2 be Complex; :: thesis: for g being complex-valued Function st g <> {} & g is non-empty & g (#) c1 = g (#) c2 holds
c1 = c2
let g be complex-valued Function; :: thesis: ( g <> {} & g is non-empty & g (#) c1 = g (#) c2 implies c1 = c2 )
assume that
A1: g <> {} and
A2: g is non-empty and
A3: g (#) c1 = g (#) c2 ; :: thesis: c1 = c2
consider x being object such that
A4: x in dom g by A1, XBOOLE_0:def 1;
g . x in rng g by A4, FUNCT_1:def 3;
then A5: g . x <> {} by A2, RELAT_1:def 9;
( (g (#) c1) . x = (g . x) * c1 & (g (#) c2) . x = (g . x) * c2 ) by VALUED_1:6;
hence c1 = c2 by A3, A5, XCMPLX_1:5; :: thesis: verum