reserve i,j,k for Nat;

theorem Th1:
  dom (Euclid-Algorithm qua Function) = 5
proof
  dom IN3 = { 3 } & dom IN4 = { 4 };
  then
A1: dom EA3 = { 3 } \/ { 4 } by FUNCT_4:def 1
    .= { 3,4 } by ENUMSET1:1;
A2: dom IN1 = { 1 };
  dom IN2 = { 2 };
  then dom EA2 = { 2 } \/ { 3,4 } by A1,FUNCT_4:def 1
    .= { 2,3,4 } by ENUMSET1:2;
  then
A3: dom EA1 = { 1 } \/ { 2,3,4 } by A2,FUNCT_4:def 1
    .= { 1,2,3,4 } by ENUMSET1:4;
  dom IN0 = { 0 };
  then dom EA0 = { 0 } \/ { 1,2,3,4 } by A3,FUNCT_4:def 1
    .= 5 by CARD_1:53,ENUMSET1:7;
  hence thesis;
end;
