theorem Th17:
  i <> k & i in dom f implies ((f,i):=(k,r)).i = k
proof
  assume that
A1: i <> k and
A2: i in dom f;
  set fik = (f,i):=k;
  thus ((f,i):=(k,r)).i =fik.i by A1,FUNCT_7:32
    .=k by A2,FUNCT_7:31;
end;
