
theorem T4a:
for n being Ordinal
for R being non degenerated Ring
for a,b being Element of R holds a|(n,R) + b|(n,R) = (a + b)|(n,R)
proof
let n be Ordinal, L be non degenerated Ring; let a,b be Element of L;
set p = (a+b)|(n,L);
A2: now let x be object;
    assume x in dom p;
    then reconsider i = x as Element of Bags n;
    per cases;
    suppose H: i = EmptyBag n;
      thus (a|(n,L) + b|(n,L)).x
         = ((a|(n,L)).i) + ((b|(n,L)).i) by POLYNOM1:15
        .= a + ((b|(n,L)).i) by H,POLYNOM7:18
        .= a + b by H,POLYNOM7:18
        .= p.x by H,POLYNOM7:18;
      end;
    suppose H: i <> EmptyBag n;
      thus (a|(n,L) + b|(n,L)).x
         = ((a|(n,L)).i) + ((b|(n,L)).i) by POLYNOM1:15
        .= 0.L + ((b|(n,L)).i) by H,POLYNOM7:18
        .= 0.L by H,POLYNOM7:18
        .= p.x by H,POLYNOM7:18;
      end;
    end;
dom p = Bags n by FUNCT_2:def 1;
hence thesis by A2;
end;
