reserve x, x1, x2, y, z, X9 for set,
  X, Y for finite set,
  n, k, m for Nat,
  f for Function;
reserve F,Ch for Function;

theorem Th32:
 for x1,x2 being object
  for Ch1,Ch2 be Function st Ch1"{x1}=Ch2"{x2} holds Intersection(
  F,Ch1,x1)=Intersection(F,Ch2,x2)
proof let x1,x2 be object;
  let Ch1,Ch2 be Function such that
A1: Ch1"{x1}=Ch2"{x2};
  thus Intersection(F,Ch1,x1)c=Intersection(F,Ch2,x2)
  proof
    let z be object such that
A2: z in Intersection(F,Ch1,x1);
    now
      let x such that
A3:   x in dom Ch2 and
A4:   Ch2.x=x2;
      Ch2.x in {x2} by A4,TARSKI:def 1;
      then
A5:   x in Ch1"{x1} by A1,A3,FUNCT_1:def 7;
      then Ch1.x in {x1} by FUNCT_1:def 7;
      then
A6:   Ch1.x=x1 by TARSKI:def 1;
      x in dom Ch1 by A5,FUNCT_1:def 7;
      hence z in F.x by A2,A6,Def2;
    end;
    hence thesis by A2,Def2;
  end;
  thus Intersection(F,Ch2,x2)c=Intersection(F,Ch1,x1)
  proof
    let z be object such that
A7: z in Intersection(F,Ch2,x2);
    now
      let x such that
A8:   x in dom Ch1 and
A9:   Ch1.x=x1;
      Ch1.x in {x1} by A9,TARSKI:def 1;
      then
A10:  x in Ch2"{x2} by A1,A8,FUNCT_1:def 7;
      then Ch2.x in {x2} by FUNCT_1:def 7;
      then
A11:  Ch2.x=x2 by TARSKI:def 1;
      x in dom Ch2 by A10,FUNCT_1:def 7;
      hence z in F.x by A7,A11,Def2;
    end;
    hence thesis by A7,Def2;
  end;
end;
