
theorem
  for M1,M2 being Matrix of COMPLEX for a being Complex st len M1
  =len M2 & width M1=width M2 holds a*(M1 + M2) = a*M1 + a*M2
proof
  let M1,M2 be Matrix of COMPLEX;
  let a be Complex;
  assume
A1: len M1=len M2 & width M1=width M2;
  a in COMPLEX by XCMPLX_0:def 2;
  then reconsider ea=a as Element of F_Complex by COMPLFLD:def 1;
A2: a*M1 + a*M2 = Field2COMPLEX ((COMPLEX2Field (Field2COMPLEX (ea*(
  COMPLEX2Field M1)))+(COMPLEX2Field (a*M2)))) by Def7
    .= Field2COMPLEX (((ea*(COMPLEX2Field M1))+(COMPLEX2Field (Field2COMPLEX
  (ea*(COMPLEX2Field M2)))))) by Def7
    .= Field2COMPLEX ((ea*(COMPLEX2Field M1)+(ea*(COMPLEX2Field M2))));
  a*(M1 + M2) = Field2COMPLEX (ea*(COMPLEX2Field (M1+M2))) by Def7
    .= Field2COMPLEX (ea*(COMPLEX2Field M1)+ea*(COMPLEX2Field M2)) by A1,Th20;
  hence thesis by A2;
end;
