reserve n,i,j,k,l for Nat;
reserve D for non empty set;
reserve c,d for Element of D;
reserve p,q,q9,r for FinSequence of D;
reserve RAS for MidSp-like non empty ReperAlgebraStr over n+2;
reserve a,b,d,pii,p9i for Point of RAS;
reserve p,q for Tuple of (n+1),RAS;
reserve m for Nat of n;
reserve W for ATLAS of RAS;
reserve v for Vector of W;
reserve x,y for Tuple of (n+1),W;
reserve RAS for ReperAlgebra of n;
reserve a,b,pm,p9m,p99m for Point of RAS;
reserve p for Tuple of (n+1),RAS;
reserve W for ATLAS of RAS;
reserve v for Vector of W;
reserve x for Tuple of (n+1),W;

theorem
  RAS has_property_of_zero_in m iff for x holds Phi((x+*(m,0.W))) = 0.W
proof
  thus RAS has_property_of_zero_in m implies for x holds Phi((x+*(m,0.W))) =
  0.W
  proof
    set a = the Point of RAS;
    assume
A1: RAS has_property_of_zero_in m;
    set b = (a,(0.W)).W;
    let x;
    set p9 = ((a,x).W)+*(m,a);
A2: b = a by MIDSP_2:34;
    then
A3: (a,((x+*(m,0.W)))).W = p9 by Th26;
    *'(a,p9) = b by A1,A2;
    hence thesis by A3,Th24;
  end;
  thus (for x holds Phi((x+*(m,0.W))) = 0.W) implies RAS
  has_property_of_zero_in m
  proof
    assume
A4: for x holds Phi((x+*(m,0.W))) = 0.W;
    for a,p holds *'(a,(p+*(m,a))) = a
    proof
      let a,p;
      set v = W.(a,a);
      set x9 = ((W.(a,p))+*(m,0.W));
      v = 0.W by MIDSP_2:33;
      then W.(a,((p+*(m,a)))) = x9 & Phi(x9) = v by A4,Th25;
      hence thesis by Th23;
    end;
    hence thesis;
  end;
end;
