reserve a,b,c,d,e,x,r for Real,
  A for non empty closed_interval Subset of REAL,
  f,g for PartFunc of REAL,REAL;

theorem Th1: ::: XREAL_0
  a <= b & c <= d & a+c = b+d implies a = b & c = d
proof
  assume that
A1: a <=b & c <=d and
A2: a+c =b+d;
  assume not (a=b & c =d);
  then a < b or c < d by A1,XXREAL_0:1;
  hence contradiction by A1,A2,XREAL_1:8;
end;
