reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;

theorem Th6:
  for A,B being Matrix of REAL st len A=len B & width A=width B
    holds len (A-B) = len A & width (A-B)=width A & 
  for i,j holds [i,j] in Indices A implies 
    (A-B)*(i,j) = A*(i,j) - B*(i,j)
proof
  let A,B be Matrix of REAL;
  assume
A1: len A=len B & width A=width B;
  thus len (A - B) =len (MXF2MXR ((MXR2MXF A)+-(MXR2MXF B))) by MATRIX_4:def 1
    .=len A by MATRIX_3:def 3;
  thus width (A - B) =width (MXF2MXR ((MXR2MXF A)+-(MXR2MXF B))) by
MATRIX_4:def 1
    .=width A by MATRIX_3:def 3;
  thus thesis by A1,MATRIX10:3;
end;
