reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem
  for A,B,C being non empty set, f,g being Function of A,[:B,C:] st
  pr1(B,C)*f = pr1(B,C)*g & pr2(B,C)*f = pr2(B,C)*g holds f = g
proof
  let A,B,C be non empty set, f,g be Function of A,[:B,C:] such that
A1: pr1(B,C)*f = pr1(B,C)*g & pr2(B,C)*f = pr2(B,C)*g;
  now
    let a be Element of A;
    consider b1 being Element of B, c1 being Element of C such that
A2: f.a = [b1,c1] by DOMAIN_1:1;
    consider b2 being Element of B, c2 being Element of C such that
A3: g.a = [b2,c2] by DOMAIN_1:1;
A4: pr1(B,C).(g.a) = (pr1(B,C)*g).a by FUNCT_2:15;
A5: pr1(B,C).(f.a) = (pr1(B,C)*f).a & pr2(B,C).(f.a) = (pr2(B,C)*f).a by
FUNCT_2:15;
A6: pr2(B,C).(b1,c1) = c1 & pr2(B,C).(b2,c2) = c2 by Def5;
    pr1(B,C).(b1,c1) = b1 & pr1(B,C).(b2,c2) = b2 by Def4;
    hence f.a = g.a by A1,A2,A3,A6,A5,A4,FUNCT_2:15;
  end;
  hence thesis;
end;
