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

theorem lmADD:
  dom (Rseq1+Rseq2) = [:NAT,NAT:] & dom (Rseq1-Rseq2) = [:NAT,NAT:] &
  (for n,m being Nat holds
   (Rseq1+Rseq2).(n,m) = Rseq1.(n,m) + Rseq2.(n,m)) &
  (for n,m being Nat holds
   (Rseq1-Rseq2).(n,m) = Rseq1.(n,m) - Rseq2.(n,m))
proof
   thus A1: dom(Rseq1+Rseq2) = [:NAT,NAT:]
          & dom(Rseq1-Rseq2) = [:NAT,NAT:] by FUNCT_2:def 1;
   hereby let n,m be Nat;
    n in NAT & m in NAT by ORDINAL1:def 12;
    hence (Rseq1+Rseq2).(n,m) = Rseq1.(n,m) + Rseq2.(n,m)
      by A1,VALUED_1:def 1,ZFMISC_1:87;
   end;
   hereby let n,m be Nat;
    n in NAT & m in NAT by ORDINAL1:def 12;
    hence (Rseq1-Rseq2).(n,m) = Rseq1.(n,m) - Rseq2.(n,m)
      by A1,VALUED_1:13,ZFMISC_1:87;
   end;
end;
