reserve x,y for object,
        D,D1,D2 for non empty set,
        i,j,k,m,n for Nat,
        f,g for FinSequence of D*,
        f1 for FinSequence of D1*,
        f2 for FinSequence of D2*;
reserve f for complex-valued Function,
        g,h for complex-valued FinSequence;

theorem Th42:
 for f be Function-yielding Function,g be FinSequence, x,y be object
   holds (g*.f)_(x,y) = g.f_(x,y)
proof
 let f be Function-yielding Function,g be FinSequence,x,y be object;
 A1:(g*.f).x = g*(f.x) by Th41;
 per cases by A1,FUNCT_1:11;
 suppose y in dom ((g*.f).x);
   hence thesis by A1,FUNCT_1:12;
 end;
 suppose not y in dom (f.x);
   then not y in dom ((g*.f).x) & f.x.y = {} by A1,FUNCT_1:11,def 2;
   then not f.x.y in dom g & (g*.f).x .y = {} by FINSEQ_3:25,FUNCT_1:def 2;
   hence thesis by FUNCT_1:def 2;
 end;
 suppose not (f.x.y) in dom g;
   then not y in dom ((g*.f).x) & g.(f.x .y) = {} by A1,FUNCT_1:11,def 2;
   hence thesis by FUNCT_1:def 2;
 end;
end;
