reserve X,Y for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,h for Function;

theorem Th9:
  f is one-to-one iff for x1,x2,y st [x1,y] in f & [x2,y] in f holds x1 = x2
proof
  thus f is one-to-one implies for x1,x2,y st [x1,y] in f & [x2,y] in f holds
  x1 = x2
  proof
    assume
A1: f is one-to-one;
    let x1,x2,y;
    assume
A2: [x1,y] in f & [x2,y] in f;
    then
A3: f.x1 = y & f.x2 = y by FUNCT_1:1;
    x1 in dom f & x2 in dom f by A2,FUNCT_1:1;
    hence thesis by A1,A3;
  end;
  assume
A4: for x1,x2,y st [x1,y] in f & [x2,y] in f holds x1 = x2;
  let x1,x2;
  assume x1 in dom f & x2 in dom f & f.x1 = f.x2;
  then [x1,f.x1] in f & [x2,f.x1] in f by FUNCT_1:1;
  hence thesis by A4;
end;
