reserve i,j for Nat;

theorem Th63:
  for x being FinSequence of REAL, A,B being Matrix of REAL st len
A=len B & width A=width B & width A=len x & len x>0 holds (A+B)*x=A*x
  + B*x
proof
  let x be FinSequence of REAL,A,B be Matrix of REAL;
  assume that
A1: len A=len B & width A=width B and
A2: width A=len x and
A4: len x>0;
A5: len ColVec2Mx x=len x by A4,Def9;
  then
A6: len (A*(ColVec2Mx x))=len A by A2,MATRIX_3:def 4
    .=len (B*(ColVec2Mx x)) by A1,A2,A5,MATRIX_3:def 4;
A7: width (A*(ColVec2Mx x))=width (ColVec2Mx x) by A2,A5,MATRIX_3:def 4
    .=1 by A4,Def9;
  thus (A+B)*x = Col(A*(ColVec2Mx x)+B*(ColVec2Mx x),1) by A1,A2,A5,
MATRIX_4:63
    .=A*x +B*x by A6,A7,Th54;
end;
