
theorem Th1:
  for g being Function, x being set st dom g = {x} holds g = x .--> g.x
proof
  let g be Function, x be set such that
A1: dom g = {x};
  now
    let a,b be object;
A2: dom (x .-->g.x) = {x};
    hereby
      assume
A3:   [a,b] in g;
      then
A4:   a in {x} by A1,FUNCT_1:1;
      then
A5:   a = x by TARSKI:def 1;
      then b = g.x by A3,FUNCT_1:1;
      then (x.-->g.x).a = b by A5,FUNCOP_1:72;
      hence [a,b] in x.-->g.x by A2,A4,FUNCT_1:1;
    end;
    assume
A6: [a,b] in x.-->g.x;
    then
A7: a in {x} by A2,FUNCT_1:1;
    then
A8: a = x by TARSKI:def 1;
    b = (x.-->g.x).a by A6,FUNCT_1:1
      .= g.a by A8,FUNCOP_1:72;
    hence [a,b] in g by A1,A7,FUNCT_1:1;
  end;
  hence thesis by RELAT_1:def 2;
end;
