reserve x,y,z for object, X for set, I for non empty set, i,j for Element of I,
    M0 for multMagma-yielding Function,
    M for non empty multMagma-yielding Function,
    M1, M2, M3 for non empty multMagma,
    G for Group-like multMagma-Family of I,
    H for Group-like associative multMagma-Family of I;

theorem Th14:
  FreeAtoms(<*M1,M2,M3*>) = [: {1}, the carrier of M1 :] \/
    [: {2}, the carrier of M2 :] \/ [: {3}, the carrier of M3 :]
proof
  set S1 = [: the carrier of M1, {1} :], S2 = [: the carrier of M2, {2} :];
  set S3 = [: the carrier of M3, {3} :], T1 = [: {1}, the carrier of M1 :];
  set T2 = [: {2}, the carrier of M2 :], T3 = [: {3}, the carrier of M3 :];
  Union disjoin Carrier <* M1, M2, M3 *>
     = Union disjoin <* the carrier of M1, the carrier of M2,
         the carrier of M3 *> by PRALG_1:19
    .= Union <* S1, S2, S3 *> by FINSEQ_3:162
    .= union rng <* S1, S2, S3 *> by CARD_3:def 4
    .= union {S1, S2, S3} by FINSEQ_2:128
    .= union ({S1,S2}\/{S3}) by ENUMSET1:3
    .= (union {S1, S2}) \/ union {S3} by ZFMISC_1:78
    .= S1 \/ S2 \/ union {S3} by ZFMISC_1:75
    .= S1 \/ S2 \/ S3 by ZFMISC_1:25
    .= T1~ \/ S2 \/ S3 by SYSREL:5
    .= T1~ \/ T2~ \/ S3 by SYSREL:5
    .= (T1 \/ T2)~ \/ S3 by RELAT_1:23
    .= (T1 \/ T2)~ \/ T3~ by SYSREL:5
    .= (T1 \/ T2 \/ T3)~ by RELAT_1:23;
  hence thesis;
end;
