let i, j be Nat; :: thesis: for M1, M2 being Matrix of COMPLEX st [i,j] in Indices M1 holds
(M1 + M2) * (i,j) = (M1 * (i,j)) + (M2 * (i,j))
let M1, M2 be Matrix of COMPLEX; :: thesis: ( [i,j] in Indices M1 implies (M1 + M2) * (i,j) = (M1 * (i,j)) + (M2 * (i,j)) )
A1: COMPLEX2Field (M1 + M2) = COMPLEX2Field (Field2COMPLEX ((COMPLEX2Field M1) + (COMPLEX2Field M2))) by MATRIX_5:def 3
.= (COMPLEX2Field M1) + (COMPLEX2Field M2) by MATRIX_5:6 ;
reconsider m1 = COMPLEX2Field M1, m2 = COMPLEX2Field M2 as Matrix of COMPLEX by COMPLFLD:def 1;
set m = COMPLEX2Field (M1 + M2);
reconsider m9 = COMPLEX2Field (M1 + M2) as Matrix of COMPLEX by COMPLFLD:def 1;
A2: M1 * (i,j) = m1 * (i,j) by MATRIX_5:def 1
.= (COMPLEX2Field M1) * (i,j) by COMPLFLD:def 1 ;
assume [i,j] in Indices M1 ; :: thesis: (M1 + M2) * (i,j) = (M1 * (i,j)) + (M2 * (i,j))
then A3: [i,j] in Indices (COMPLEX2Field M1) by MATRIX_5:def 1;
A4: M2 * (i,j) = m2 * (i,j) by MATRIX_5:def 1
.= (COMPLEX2Field M2) * (i,j) by COMPLFLD:def 1 ;
(M1 + M2) * (i,j) = m9 * (i,j) by MATRIX_5:def 1
.= (COMPLEX2Field (M1 + M2)) * (i,j) by COMPLFLD:def 1
.= ((COMPLEX2Field M1) * (i,j)) + ((COMPLEX2Field M2) * (i,j)) by A1, A3, MATRIX_3:def 3 ;
hence (M1 + M2) * (i,j) = (M1 * (i,j)) + (M2 * (i,j)) by A2, A4; :: thesis: verum