 reserve Rseq, Rseq1, Rseq2 for Function of [:NAT,NAT:],REAL;

theorem
  for C,D being non empty set, F1,F2 being Function of [:C,D:],REAL,
      c being Element of C holds
  ProjMap1(F1+F2,c) = ProjMap1(F1,c) + ProjMap1(F2,c)
proof
   let C,D be non empty set;
   let F1,F2 be Function of [:C,D:],REAL;
   let c be Element of C;
   dom ProjMap1(F1+F2,c) = D
 & dom ProjMap1(F1,c) = D & dom ProjMap1(F2,c) = D by FUNCT_2:def 1; then
A2:dom ProjMap1(F1+F2,c) = dom ProjMap1(F1,c) /\ dom ProjMap1(F2,c);
   for d being object st d in dom ProjMap1(F1+F2,c) holds
    ProjMap1(F1+F2,c).d = ProjMap1(F1,c).d + ProjMap1(F2,c).d
   proof
    let d be object;
    assume A3: d in dom ProjMap1(F1+F2,c); then
A4: ProjMap1(F1+F2,c).d = (F1+F2).(c,d) & ProjMap1(F1,c).d = F1.(c,d)
  & ProjMap1(F2,c).d = F2.(c,d) by MESFUNC9:def 6;
    reconsider d1=d as Element of D by A3;
    [c,d] in [:C,D:] by A3,ZFMISC_1:def 2; then
    [c,d] in dom(F1+F2) by FUNCT_2:def 1;
    hence thesis by A4,VALUED_1:def 1;
   end;
   hence thesis by A2,VALUED_1:def 1;
end;
