reserve A for non empty closed_interval Subset of REAL;
reserve rho for Function of A,REAL;
reserve u for PartFunc of REAL,REAL;
reserve T for DivSequence of A;
reserve S for middle_volume_Sequence of rho,u,T;
reserve k for Nat;

theorem Lm7A:
  for A,B be non empty closed_interval Subset of REAL,
      rho,rho1,rho2 be Function of A,REAL
    st B c= A & rho = rho1 - rho2
   holds vol(B,rho) = vol(B,rho1) - vol(B,rho2)
proof
  let A,B be non empty closed_interval Subset of REAL,
      rho,rho1,rho2 be Function of A,REAL;
  assume AS: B c= A & rho = rho1 - rho2; then
A1: rho = rho1 + (-rho2) by VALUED_1:def 9;
  set x1=upper_bound B;
  set x2=lower_bound B;
  thus vol(B,rho) = vol(B,rho1) + vol(B,-rho2) by AS,A1,Lm6A
   .= vol(B,rho1) + ((-rho2).x1 - (-rho2).x2) by Defvol
   .= vol(B,rho1) + (-rho2.x1 - (-rho2).x2) by VALUED_1:8
   .= vol(B,rho1) + (-rho2.x1 - -rho2.x2) by VALUED_1:8
   .= vol(B,rho1) - (rho2.x1 - rho2.x2)
   .= vol(B,rho1) - vol(B,rho2) by Defvol;
end;
