reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;
reserve A,B for set;
reserve x,y,i,j,k for object;
reserve x for set;
reserve x for object;
reserve A1,A2,B1,B2 for non empty set,
  f for Function of A1,B1,
  g for Function of A2,B2,
  Y1 for non empty Subset of A1,
  Y2 for non empty Subset of A2;
reserve a,b,c,x,y,z,w,d for object;

theorem
  for f being Function st dom f = {a,b,c} & f.a = x & f.b = y & f.c = z holds
  f = (a,b,c) --> (x,y,z)
  proof
    let f be Function such that
A1: dom f = {a,b,c} and
A2: f.a = x & f.b = y & f.c = z;
    set g = (a,b,c) --> (x,y,z);
    thus dom f = dom g by A1,Th128;
    let k be object;
    assume k in dom f;
    then
A3: k = a or k = b or k = c by A1,ENUMSET1:def 1;
    per cases;
    suppose a,b,c are_mutually_distinct;
      hence thesis by A2,A3,Th134,Th135;
    end;
    suppose
A4:     a = b & a <> c;
        then g = (a,c) --> (y,z) by Th81;
        hence thesis by A4,A2,A3,Th63;
    end;
    suppose
A5:     a = c;
        per cases;
        suppose
A6:       a <> b;
          then g = (a,b) --> (z,y) by A5,Th132;
          hence thesis by A5,A2,A3,A6,Th63;
        end;
        suppose a = b;
          hence thesis by A5,A2,A3,FUNCOP_1:72;
        end;
      end;
      suppose
A7:     b = c & a <> c;
        thus thesis by A7,A2,A3,Th63;
    end;
  end;
