reserve A for set;
reserve X,Y,Z for set,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve u for UnOp of A,
  o,o9 for BinOp of A,
  a,b,c,e,e1,e2 for Element of A;

theorem
  for f being Function st dom f = [:X,Y:] holds f is constant iff for x1
  ,x2,y1,y2 st x1 in X & x2 in X & y1 in Y & y2 in Y holds f.(x1,y1)=f.(x2,y2)
proof
  let f be Function such that
A1: dom f = [:X,Y:];
  hereby
    assume
A2: f is constant;
    let x1,x2,y1,y2;
    assume x1 in X & x2 in X & y1 in Y & y2 in Y;
    then [x1,y1] in [:X,Y:] & [x2,y2] in [:X,Y:] by ZFMISC_1:87;
    hence f.(x1,y1)=f.(x2,y2) by A1,A2;
  end;
  assume
A3: for x1,x2,y1,y2 st x1 in X & x2 in X & y1 in Y & y2 in Y holds f.(x1
  ,y1)=f.(x2,y2);
  let x,y be object;
  assume x in dom f;
  then consider x1,y1 being object such that
A4: x1 in X & y1 in Y and
A5: x = [x1,y1] by A1,ZFMISC_1:84;
  assume y in dom f;
  then consider x2,y2 being object such that
A6: x2 in X & y2 in Y and
A7: y = [x2,y2] by A1,ZFMISC_1:84;
  thus f.x = f.(x1,y1) by A5
    .= f.(x2,y2) by A3,A4,A6
    .= f.y by A7;
end;
