reserve a,b,c,d for Real;

theorem
  P[01](0,1,(#)(0,1),(0,1)(#)) = id Closed-Interval-TSpace(0,1)
proof
  for x being Point of Closed-Interval-TSpace(0,1) holds P[01](0,1,(#)(0,1
  ),(0,1)(#)).x = x
  proof
    let x be Point of Closed-Interval-TSpace(0,1);
    reconsider y = x as Real;
    (#)(0,1)=0 & (0,1)(#) = 1 by Def1,Def2;
    hence P[01](0,1,(#)(0,1),(0,1)(#)).x = ((1-y)*0 + (y-0)*1)/(1-0) by Def4
      .= x;
  end;
  hence thesis by FUNCT_2:124;
end;
