reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;
reserve X for non empty set,
  x for Element of X;
reserve F for Part-Family of X;
reserve e,u,v for object, E,X,Y,X1 for set;

theorem Th49:
  for X1 being Subset of X, Y1 being Subset of Y st [:X1,Y1:] <>
  {} holds pr1(X,Y).:[:X1,Y1:] = X1 & pr2(X,Y).:[:X1,Y1:] = Y1
proof
  let X1 be Subset of X, Y1 be Subset of Y;
  assume
A1: [:X1,Y1:] <> {};
  then
A2: X1 <> {};
A3: Y1 <> {} by A1;
A4: X <> {} by A2;
  now
    set y = the Element of Y1;
    let x be object;
    thus x in pr1(X,Y).:[:X1,Y1:] implies x in X1
    proof
      assume x in pr1(X,Y).:[:X1,Y1:];
      then consider u such that
A5:   u in [:X,Y:] and
A6:   u in [:X1,Y1:] & x = pr1(X,Y).u by FUNCT_2:64;
A7:   u`2 in Y by A5,MCART_1:10;
      u`1 in X1 & x = pr1(X,Y).(u`1,u`2) by A6,MCART_1:10,21;
      hence thesis by A7,FUNCT_3:def 4;
    end;
    assume
A8: x in X1;
    y in Y by A3,TARSKI:def 3;
    then
A9: x = pr1(X,Y).(x,y) by A8,FUNCT_3:def 4;
    [x,y] in [:X1,Y1:] by A3,A8,ZFMISC_1:87;
    hence x in pr1(X,Y).:[:X1,Y1:] by A4,A9,FUNCT_2:35;
  end;
  hence pr1(X,Y).:[:X1,Y1:] = X1 by TARSKI:2;
A10: Y <> {} by A3;
  now
    set x = the Element of X1;
    let y be object;
    thus y in pr2(X,Y).:[:X1,Y1:] implies y in Y1
    proof
      assume y in pr2(X,Y).:[:X1,Y1:];
      then consider u such that
A11:  u in [:X,Y:] and
A12:  u in [:X1,Y1:] & y = pr2(X,Y).u by FUNCT_2:64;
A13:  u`1 in X by A11,MCART_1:10;
      u`2 in Y1 & y = pr2(X,Y).(u`1,u`2) by A12,MCART_1:10,21;
      hence thesis by A13,FUNCT_3:def 5;
    end;
    assume
A14: y in Y1;
    x in X by A2,TARSKI:def 3;
    then
A15: y = pr2(X,Y).(x,y) by A14,FUNCT_3:def 5;
    [x,y] in [:X1,Y1:] by A2,A14,ZFMISC_1:87;
    hence y in pr2(X,Y).:[:X1,Y1:] by A10,A15,FUNCT_2:35;
  end;
  hence thesis by TARSKI:2;
end;
